MTH120 College Algebra Chapter 2.2

 2.2 Linear Equations in One Variable

Solving linear equations in one variable is a fundamental skill in algebra. A linear equation is an equation of the form $��+�=0$, where $�$ and $�$ are constants, and $�$ is the variable you're solving for. Here are the general steps to solve a linear equation in one variable:

Step 1: Isolate the Variable Term:

• Your goal is to isolate the variable $�$ on one side of the equation.
• To do this, perform the opposite operations to both sides of the equation until $�$ is alone on one side.

Step 2: Simplify Both Sides:

• Simplify each side of the equation by combining like terms and performing arithmetic operations.

Step 3: Solve for $�$:

• Once $�$ is isolated, you can determine its value by solving the equation.

Step 4: Check Your Solution:

• Always check your solution by plugging it back into the original equation to ensure it satisfies the equation.

Let's work through a few examples:

Example 1: Solve for $�$ in the equation $3�+5=11$.

Step 1: Isolate the Variable Term:

• Subtract 5 from both sides to isolate the variable: $3�+5-5=11-5$ $3�=6$

Step 2: Simplify Both Sides:

• There are no like terms to simplify in this step.

Step 3: Solve for $�$:

• Divide both sides by 3 to solve for $�$: $\frac{3�}{3}=\frac{6}{3}$ $�=2$

Step 4: Check Your Solution:

• Plug $�=2$ back into the original equation: $3(2)+5=11$ $6+5=11$ $11=11$
• The equation is true, so the solution is correct: $�=2$.

Example 2: Solve for $�$ in the equation $\frac{2}{3}�-4=7$.

Step 1: Isolate the Variable Term:

• Add 4 to both sides to isolate the variable: $\frac{2}{3}�-4+4=7+4$ $\frac{2}{3}�=11$

Step 2: Simplify Both Sides:

• There are no like terms to simplify in this step.

Step 3: Solve for $�$:

• Multiply both sides by $\frac{3}{2}$ to solve for $�$: $\frac{3}{2}\cdot \frac{2}{3}�=\frac{11}{1}\cdot \frac{3}{2}$ $�=\frac{33}{2}$

Step 4: Check Your Solution:

• Plug $�=\frac{33}{2}$ back into the original equation: $\frac{2}{3}\cdot \frac{33}{2}-4=7$ $11-4=7$ $7=7$
• The equation is true, so the solution is correct: $�=\frac{33}{2}$.

These examples demonstrate the process of solving linear equations in one variable by isolating the variable term and solving for $�$. Always check your solution to ensure its accuracy.

Let's solve a linear equation in one variable step by step.

Example: Solve for $�$ in the equation $2�+3=7$.

Step 1: Isolate the Variable Term:

• Your goal is to isolate the variable term, which is $2�$, on one side of the equation.
• To do this, subtract 3 from both sides of the equation to move the constant term to the other side: $2�+3-3=7-3$ $2�=4$

Step 2: Simplify Both Sides:

• There are no like terms to simplify further in this step.

Step 3: Solve for $�$:

• Now that $2�$ is isolated on the left side, you can solve for $�$ by dividing both sides by 2: $\frac{2�}{2}=\frac{4}{2}$ $�=2$

Step 4: Check Your Solution:

• It's always a good practice to check your solution by plugging it back into the original equation to make sure it satisfies the equation: $2(2)+3=7$ $4+3=7$ $7=7$
• The equation is true, so the solution is correct: $�=2$.

So, the solution to the equation $2�+3=7$ is $�=2$.

When the variable appears on both sides of an equation, you can still solve it algebraically by performing a sequence of operations to isolate the variable on one side. Here's a step-by-step guide:

Example: Solve for $�$ in the equation $3�-5=2�+7$.

Step 1: Identify the Variable Terms:

• In this equation, you have variable terms on both sides: $3�$ on the left side and $2�$ on the right side.

Step 2: Isolate the Variable Terms:

• Your goal is to get all the variable terms on one side and constants on the other side. To do this, you need to move $2�$ from the right side to the left side. To achieve this, subtract $2�$ from both sides: $3�-5-2�=2�+7-2�$ $�-5=7$

Step 3: Simplify Both Sides:

• Now, simplify both sides of the equation by combining like terms and performing arithmetic operations: $�-5+5=7+5$ $�=12$

Step 4: Check Your Solution:

• Always check your solution by substituting it back into the original equation to make sure it satisfies the equation: $3(12)-5=2(12)+7$ $36-5=24+7$ $31=31$
• The equation is true, so the solution is correct: $�=12$.

So, the solution to the equation $3�-5=2�+7$ is $�=12$.

In this example, we moved the variable terms to one side and the constants to the other side by performing operations (in this case, subtracting $2�$ from both sides) to isolate the variable term, and then we simplified the equation to find the solution.

Solving a rational equation involves finding the values of the variable that satisfy the equation. Rational equations are equations that contain fractions with variables in the numerator and/or denominator. Here's a step-by-step guide on how to solve a rational equation:

Example: Solve for $�$ in the equation $\frac{�}{3}-\frac{2}{�}=1$.

Step 1: Find a Common Denominator:

• To eliminate fractions, find a common denominator for all the terms. In this case, the common denominator is $3�$.

Step 2: Multiply Both Sides by the Common Denominator:

• Multiply both sides of the equation by $3�$ to clear the fractions: $3�(\frac{�}{3}-\frac{2}{�})=3�\cdot 1$ ${�}^2-6=3�$

Step 3: Move All Terms to One Side:

• Rearrange the equation to have all terms on one side, typically with 0 on the other side: ${�}^2-3�-6=0$

Step 4: Factor or Use the Quadratic Formula:

• Attempt to factor the quadratic equation. In this case, it can be factored as $(�-3)(�+2)=0$.
• Set each factor equal to 0: $�-3=0\text{or}�+2=0$

Step 5: Solve for $�$:

• Solve each equation separately:
  • $�-3=0⟹�=3$
  • $�+2=0⟹�=-2$

Step 6: Check Your Solutions:

• Always check your solutions by substituting them back into the original equation to ensure they satisfy the equation:

  • For $�=3$: $\frac{3}{3}-\frac{2}{3}=1$ $1-\frac{2}{3}=1$ $\frac{3}{3}-\frac{2}{3}=1$ $1-\frac{2}{3}=1$ $\frac{3}{3}-\frac{2}{3}=1$ $1-\frac{2}{3}=1$
  • For $�=-2$: $\frac{-2}{3}-\frac{2}{-2}=1$ $-\frac{2}{3}+1=1$ $\frac{-2}{3}-\frac{2}{-2}=1$ $-\frac{2}{3}+1=1$ $\frac{-2}{3}-\frac{2}{-2}=1$ $-\frac{2}{3}+1=1$

• Both solutions satisfy the original equation.

So, the solutions to the equation $\frac{�}{3}-\frac{2}{�}=1$ are $�=3$ and $�=-2$.

Solving a rational equation by factoring the denominator involves finding the values of the variable that satisfy the equation while considering the restrictions imposed by the denominator. Here's a step-by-step guide on how to solve a rational equation by factoring the denominator:

Example: Solve for $�$ in the equation $\frac{2}{�-1}=\frac{3}{�+2}$.

Step 1: Find the Common Denominator:

• To eliminate fractions, find a common denominator for both fractions. In this case, the common denominator is $(�-1)(�+2)$, which combines the two denominators.

Step 2: Rewrite the Fractions with the Common Denominator:

• Rewrite both fractions with the common denominator: $\frac{2}{�-1}\cdot \frac{(�+2)}{(�+2)}=\frac{3}{�+2}\cdot \frac{(�-1)}{(�-1)}$

Step 3: Eliminate the Denominator:

• After rewriting the fractions, the denominators cancel out: $2(�+2)=3(�-1)$

Step 4: Distribute and Simplify:

• Distribute on both sides of the equation and simplify: $2�+4=3�-3$

Step 5: Move All Terms to One Side:

• Rearrange the equation to have all terms on one side, typically with 0 on the other side: $2�-3�+4+3=0$

Step 6: Simplify Further:

• Combine like terms: $-�+7=0$

Step 7: Solve for $�$:

• Isolate the variable term by subtracting 7 from both sides: $-�+7-7=0-7$ $-�=-7$

Step 8: Solve for $�$ (continued):

• Multiply both sides by -1 to solve for $�$: $-1\cdot (-�)=-1\cdot (-7)$ $�=7$

Step 9: Check Your Solution:

• Always check your solution by substituting it back into the original equation to ensure it satisfies the equation:

  • For $�=7$: $\frac{2}{7-1}=\frac{3}{7+2}$ $\frac{2}{6}=\frac{3}{9}$ $\frac{1}{3}=\frac{1}{3}$

• The solution $�=7$ satisfies the original equation.

So, the solution to the equation $\frac{2}{�-1}=\frac{3}{�+2}$ is $�=7$.

Solving rational equations with a binomial in the denominator follows similar principles to solving other rational equations. Here's a step-by-step guide on how to solve a rational equation with a binomial in the denominator:

Example: Solve for $�$ in the equation $\frac{1}{�+3}=\frac{2}{�-1}$.

Step 1: Find the Common Denominator:

• To eliminate fractions, find a common denominator for both fractions. In this case, the common denominator is $(�+3)(�-1)$, which combines the two denominators.

Step 2: Rewrite the Fractions with the Common Denominator:

• Rewrite both fractions with the common denominator: $\frac{1}{�+3}\cdot \frac{(�-1)}{(�-1)}=\frac{2}{�-1}\cdot \frac{(�+3)}{(�+3)}$

Step 3: Eliminate the Denominator:

• After rewriting the fractions, the denominators cancel out: $1(�-1)=2(�+3)$

Step 4: Distribute and Simplify:

• Distribute on both sides of the equation and simplify: $�-1=2�+6$

Step 5: Move All Terms to One Side:

• Rearrange the equation to have all terms on one side, typically with 0 on the other side: $�-2�-1-6=0$

Step 6: Simplify Further:

• Combine like terms: $-�-7=0$

Step 7: Solve for $�$:

• Isolate the variable term by adding 7 to both sides: $-�-7+7=0+7$ $-�=7$

Step 8: Solve for $�$ (continued):

• Multiply both sides by -1 to solve for $�$: $-1\cdot (-�)=-1\cdot 7$ $�=-7$

Step 9: Check Your Solution:

• Always check your solution by substituting it back into the original equation to ensure it satisfies the equation:

  • For $�=-7$: $\frac{1}{(-7)+3}=\frac{2}{(-7)-1}$ $\frac{1}{-4}=\frac{2}{-8}$ $-\frac{1}{4}=-\frac{1}{4}$

• The solution $�=-7$ satisfies the original equation.

So, the solution to the equation $\frac{1}{�+3}=\frac{2}{�-1}$ is $�=-7$.

When solving a rational equation with factored denominators, you need to consider the excluded values, which are values of the variable that would make the denominator equal to zero (and thus make the fraction undefined). Here's a step-by-step guide on how to solve such an equation:

Example: Solve for $�$ in the equation $\frac{�}{�+3}-\frac{2}{�-1}=0$.

Step 1: Find the Common Denominator:

• To eliminate fractions, find a common denominator for both fractions. In this case, the common denominator is $(�+3)(�-1)$, which combines the two denominators.

Step 2: Rewrite the Fractions with the Common Denominator:

• Rewrite both fractions with the common denominator: $\frac{�}{�+3}\cdot \frac{(�-1)}{(�-1)}-\frac{2}{�-1}\cdot \frac{(�+3)}{(�+3)}=0$

Step 3: Combine Fractions:

• Combine the two fractions with a common denominator into a single fraction: $\frac{�(�-1)-2(�+3)}{(�+3)(�-1)}=0$

Step 4: Expand and Simplify:

• Expand and simplify the numerator: ${�}^2-�-2�-6=0$ ${�}^2-3�-6=0$

Step 5: Factor the Quadratic Equation:

• Factor the quadratic equation, if possible: $(�-6)(�+1)=0$

Step 6: Solve for $�$:

• Set each factor equal to 0 and solve for $�$:
  • $�-6=0⟹�=6$
  • $�+1=0⟹�=-1$

Step 7: Identify Excluded Values:

• Excluded values are values of $�$ that make the denominators equal to zero. In this case, the excluded values are $�=-3$ (from $�+3=0$) and $�=1$ (from $�-1=0$) because these values would result in division by zero, making the fractions undefined.

Step 8: Check Your Solutions:

• Always check your solutions by substituting them back into the original equation to ensure they satisfy the equation:

  • For $�=6$: $\frac{6}{6+3}-\frac{2}{6-1}=0$ $\frac{6}{9}-\frac{2}{5}=0$ $\frac{2}{3}-\frac{2}{5}=0$ $\frac{10-6}{15}=0$ $\frac{4}{15}=0$

  • For $�=-1$: $\frac{-1}{-1+3}-\frac{2}{-1-1}=0$ $\frac{-1}{2}-\frac{2}{-2}=0$ $-\frac{1}{2}+1=0$ $\frac{2-1}{2}=0$ $\frac{1}{2}=0$

• Neither solution satisfies the original equation.

Step 9: State the Final Solution:

• After checking the solutions, you find that neither $�=6$ nor $�=-1$ satisfy the original equation. Therefore, there are no valid solutions to the equation.

In this example, you need to be cautious about excluded values and ensure that the solutions satisfy the original equation. In this case, there are no valid solutions.

To find a linear equation, you need two pieces of information: a point (x, y) on the line and the slope of the line. A linear equation can be written in the form:

$�=��+�$

Where:

• $�$ is the dependent variable (usually representing the vertical axis in the Cartesian coordinate system).
• $�$ is the independent variable (usually representing the horizontal axis).
• $�$ is the slope of the line.
• $�$ is the y-intercept, which is the value of $�$ when $�=0$.

Here's how to find a linear equation given a point and the slope:

Step 1: Identify the Slope (m):

• You'll need to know the slope of the line. If you have two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ on the line, you can calculate the slope as: $�=\frac{{�}_2-{�}_1}{{�}_2-{�}_1}$

Step 2: Use a Point (x, y) on the Line:

• Choose a point $(�,�)$ that lies on the line. You'll use this point to find the y-intercept ($�$).

Step 3: Calculate the Y-Intercept (b):

• Use the point $(�,�)$ and the slope $�$ to calculate the y-intercept $�$ using the formula: $�=�-��$

Step 4: Write the Linear Equation:

• With the slope $�$ and y-intercept $�$, write the linear equation in the form $�=��+�$.

Let's work through an example:

Example: Find the equation of a line with a slope of 2 that passes through the point (3, 5).

Step 1: Identify the Slope (m):

• We're given the slope $�=2$.

Step 2: Use a Point (x, y) on the Line:

• We're given the point (3, 5).

Step 3: Calculate the Y-Intercept (b):

• Use the point (3, 5) and the slope $�=2$ to find the y-intercept $�$: $�=5-2\cdot 3$ $�=5-6$ $�=-1$

Step 4: Write the Linear Equation:

• Now that we have the slope $�=2$ and y-intercept $�=-1$, we can write the linear equation: $�=2�-1$

So, the equation of the line with a slope of 2 that passes through the point (3, 5) is $�=2�-1$. This equation describes a straight line with a slope of 2 and a y-intercept of -1.

The slope of a line is a measure of how steep the line is and is represented by the letter "m." It indicates the rate at which the dependent variable (usually denoted as "y") changes with respect to a change in the independent variable (usually denoted as "x") along the line.

The formula to calculate the slope ($�$) between two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ on a line is:

$�=\frac{{�}_2-{�}_1}{{�}_2-{�}_1}$

In this formula:

• ${�}_2$ and ${�}_1$ are the y-coordinates of the two points.
• ${�}_2$ and ${�}_1$ are the x-coordinates of the two points.

Here's what the slope represents:

• A positive slope ($�>0$) indicates that as $�$ increases, $�$ also increases, leading to an upward-sloping line.
• A negative slope ($�<0$) indicates that as $�$ increases, $�$ decreases, resulting in a downward-sloping line.
• A zero slope ($�=0$) implies that $�$ remains constant as $�$ changes, resulting in a horizontal line.
• An undefined slope occurs when the denominator (${�}_2-{�}_1$) is zero, meaning the line is vertical, and its slope is considered undefined.

The slope provides valuable information about the relationship between the two variables, and it's a fundamental concept in algebra and geometry. It allows you to quantify how steep a line is and make predictions about how the dependent variable changes as the independent variable changes.

To find the slope of a line given two points on the line, you can use the slope formula. The slope formula calculates the rate of change between the two points. Here are the steps to find the slope:

Step 1: Identify the Coordinates of the Two Points:

• You need the coordinates of two distinct points on the line. These points can be represented as $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$.

Step 2: Use the Slope Formula:

• Apply the slope formula to calculate the slope ($�$) between the two points: $�=\frac{{�}_2-{�}_1}{{�}_2-{�}_1}$

Step 3: Calculate the Slope:

• Plug in the coordinates of the two points into the formula and perform the arithmetic operations to find the slope.

Step 4: Interpret the Slope:

• The result from Step 3 is the slope of the line. Depending on the value of $�$:
  • If $�>0$, it indicates an upward-sloping line.
  • If $�<0$, it indicates a downward-sloping line.
  • If $�=0$, it indicates a horizontal line.
  • If the denominator (${�}_2-{�}_1$) is zero, the slope is undefined, indicating a vertical line.

Let's work through an example:

Example: Find the slope of the line passing through the points $(2,3)$ and $(5,9)$.

Step 1: Identify the Coordinates of the Two Points:

• The first point is $(2,3)$, and the second point is $(5,9)$.

Step 2: Use the Slope Formula:

• Apply the slope formula: $�=\frac{{�}_2-{�}_1}{{�}_2-{�}_1}$

Step 3: Calculate the Slope:

• Plug in the coordinates of the two points and calculate the slope: $�=\frac{9-3}{5-2}$ $�=\frac{6}{3}$ $�=2$

Step 4: Interpret the Slope:

• The slope $�=2$ indicates that the line is upward-sloping. For every 1 unit increase in $�$, $�$ increases by 2 units.

So, the slope of the line passing through the points $(2,3)$ and $(5,9)$ is $�=2$.

To identify the slope and y-intercept of a line given its equation in slope-intercept form ($�=��+�$), you can simply look at the coefficients of $�$ and the constant term ($�$) in the equation. Here's how to do it:

Slope-Intercept Form of a Line Equation: $�=��+�$

• $�$ represents the slope of the line.
• $�$ represents the y-intercept, which is the point where the line crosses the y-axis ($�=0$).

Steps:

1. Identify the Slope (m): Look at the coefficient of $�$. This coefficient is $�$, the slope of the line. If the equation is not written in slope-intercept form, you may need to rearrange it to this form.

2. Identify the Y-Intercept (b): Look at the constant term ($�$) in the equation. This constant is the y-intercept, which is the value of $�$ when $�=0$.

Here are some examples to illustrate this process:

Example 1: Identify the slope and y-intercept of the line with the equation $�=3�+2$.

• Slope (m): The coefficient of $�$ is 3, so the slope is $�=3$.
• Y-Intercept (b): The constant term is 2, so the y-intercept is $�=2$.

So, the slope of the line is 3, and the y-intercept is 2.

Example 2: Identify the slope and y-intercept of the line with the equation $2�-4�=8$.

• Rearrange the Equation: To identify the slope and y-intercept, rearrange the equation into slope-intercept form ($�=��+�$): $2�=4�+8$ $�=2�+4$

• Slope (m): The coefficient of $�$ is 2, so the slope is $�=2$.

• Y-Intercept (b): The constant term is 4, so the y-intercept is $�=4$.

So, the slope of the line is 2, and the y-intercept is 4.

In summary, you can easily identify the slope and y-intercept of a line by examining the coefficients of $�$ and the constant term in the equation when it's in slope-intercept form ($�=��+�$).

The point-slope formula is a way to write the equation of a straight line when you know a point on the line and its slope. The formula is written as:

$�-{�}_1=�(�-{�}_1)$

Where:

• $({�}_1,{�}_1)$ represents a point on the line.
• $�$ represents the slope of the line.

Here's how you can use the point-slope formula:

Step 1: Identify the Point (x1, y1):

• You need to know one point that lies on the line. This point is represented as $({�}_1,{�}_1)$.

Step 2: Determine the Slope (m):

• You also need to know the slope ($�$) of the line. The slope can be given or calculated from other information.

Step 3: Plug the Values into the Formula:

• Once you have the point $({�}_1,{�}_1)$ and the slope $�$, you can plug them into the point-slope formula: $�-{�}_1=�(�-{�}_1)$

Step 4: Simplify if Needed:

• If necessary, simplify the equation to make it more readable or to match the desired form. For example, you can solve for $�$ to put the equation in slope-intercept form ($�=��+�$).

The point-slope formula allows you to quickly write the equation of a line when you have a specific point and its slope. It's particularly useful when you don't have the y-intercept but do have other information about the line.

Let's work through an example:

Example: Write the equation of a line with a slope of 2 that passes through the point (3, 4) using the point-slope formula.

Step 1: Identify the Point (x1, y1):

• We're given the point (3, 4), which is $({�}_1,{�}_1)$.

Step 2: Determine the Slope (m):

• We're given the slope $�=2$.

Step 3: Plug the Values into the Formula:

• Use the point-slope formula: $�-4=2(�-3)$

Step 4: Simplify if Needed:

• You can simplify further if desired. In this case, let's solve for $�$: $�=2�-6+4$ $�=2�-2$

So, the equation of the line with a slope of 2 that passes through the point (3, 4) is $�=2�-2$.

To find the equation of a line given the slope ($�$) and one point (${�}_1,{�}_1$) on the line, you can use the point-slope formula:

$�-{�}_1=�(�-{�}_1)$

Here's how to use this formula:

Step 1: Identify the Point (x1, y1):

• You need to know one point $({�}_1,{�}_1)$ on the line.

Step 2: Determine the Slope (m):

• You also need to know the slope ($�$) of the line.

Step 3: Plug the Values into the Formula:

• Once you have the point $({�}_1,{�}_1)$ and the slope $�$, plug these values into the point-slope formula: $�-{�}_1=�(�-{�}_1)$

Step 4: Simplify if Needed:

• You can simplify the equation to make it more readable or to match a specific form (e.g., slope-intercept form $�=��+�$).

Let's work through an example:

Example: Find the equation of a line with a slope of 3 that passes through the point (2, 4).

Step 1: Identify the Point (x1, y1):

• We're given the point (2, 4), which is $({�}_1,{�}_1)$.

Step 2: Determine the Slope (m):

• We're given the slope $�=3$.

Step 3: Plug the Values into the Formula:

• Use the point-slope formula: $�-4=3(�-2)$

Step 4: Simplify if Needed:

• Let's simplify the equation: $�-4=3�-6$

Now, the equation is in a simplified form. If you want to express it in slope-intercept form ($�=��+�$), you can isolate $�$:

$�=3�-6+4$ $�=3�-2$

So, the equation of the line with a slope of 3 that passes through the point (2, 4) is $�=3�-2$.

To find the equation of a line passing through two given points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$, you can use the point-slope formula and follow these steps:

Step 1: Calculate the Slope (m):

• Use the coordinates of the two points to calculate the slope ($�$) of the line using the formula: $�=\frac{{�}_2-{�}_1}{{�}_2-{�}_1}$

Step 2: Choose One of the Points (x1, y1):

• Select one of the two given points $({�}_1,{�}_1)$ to use in the point-slope formula.

Step 3: Use the Point-Slope Formula:

• Apply the point-slope formula, using the slope ($�$) calculated in Step 1 and the chosen point $({�}_1,{�}_1)$: $�-{�}_1=�(�-{�}_1)$

Step 4: Simplify if Needed:

• If desired, simplify the equation to match a specific form, such as slope-intercept form ($�=��+�$).

Let's work through an example:

Example: Find the equation of the line passing through the points (2, 4) and (5, 10).

Step 1: Calculate the Slope (m):

• Calculate the slope $�$ using the coordinates of the two points: $�=\frac{10-4}{5-2}=\frac{6}{3}=2$

Step 2: Choose One of the Points (x1, y1):

• We can choose the point (2, 4) as $({�}_1,{�}_1)$.

Step 3: Use the Point-Slope Formula:

• Apply the point-slope formula with the calculated slope $�$ and the chosen point $(2,4)$: $�-4=2(�-2)$

Step 4: Simplify if Needed:

• Let's simplify the equation: $�-4=2�-4$

Now, the equation is in a simplified form. If you want to express it in slope-intercept form ($�=��+�$), you can isolate $�$:

$�=2�-4+4$ $�=2�$

So, the equation of the line passing through the points (2, 4) and (5, 10) is $�=2�$.

The standard form of a linear equation is a way to write the equation of a line in a particular, standard format. A linear equation in standard form is typically written as:

$��+��=�$

Where:

• $�$, $�$, and $�$ are integers (positive or negative whole numbers).
• $�$ is a positive integer (it can be zero).
• $�$ is also an integer, and it can be positive or negative.
• $�$ is an integer, and it can be positive or negative.

The key characteristics of the standard form of a linear equation are as follows:

1. All variables ($�$ and $�$) are on the left-hand side of the equation.
2. The coefficients $�$ and $�$ are integers.
3. The coefficients $�$ and $�$ should have no common factors other than 1. In other words, they should be relatively prime.

Converting a linear equation from another form (e.g., slope-intercept form or point-slope form) to standard form often involves algebraic manipulation. Here's an example of how to convert a linear equation from slope-intercept form to standard form:

Example: Convert the equation $�=3�-2$ to standard form.

Step 1: Write the Equation in Standard Form:

• Start with the given equation $�=3�-2$.
• Move all terms to the left-hand side: $�-3�=-2$

Step 2: Ensure Coefficients are Integers:

• In this case, the coefficients are already integers, so no additional steps are needed.

Step 3: Check for Relative Primality:

• The coefficients 1 and -3 have no common factors other than 1, so they are relatively prime.

The equation $�-3�=-2$ is now in standard form, as it meets all the criteria mentioned earlier. This format is particularly useful when working with linear equations in contexts such as systems of equations or when you need to compare coefficients.

To find the equation of a line and write it in standard form ($��+��=�$), you'll typically need either:

1. Two points through which the line passes, or
2. One point through which the line passes and the slope of the line.

Here are the steps to find the equation of a line and write it in standard form:

Method 1: Using Two Points (x1, y1) and (x2, y2):

Step 1: Calculate the Slope (m):

• Calculate the slope ($�$) of the line using the two points: $�=\frac{{�}_2-{�}_1}{{�}_2-{�}_1}$

Step 2: Use One of the Points (x1, y1):

• Choose one of the two points, say $({�}_1,{�}_1)$.

Step 3: Use the Point-Slope Formula:

• Apply the point-slope formula with the calculated slope $�$ and the chosen point $({�}_1,{�}_1)$: $�-{�}_1=�(�-{�}_1)$

Step 4: Simplify the Equation:

• Simplify the equation to make it easier to work with.

Method 2: Using One Point (x1, y1) and the Slope (m):

Step 1: Choose One Point (x1, y1):

• Choose the point through which the line passes, say $({�}_1,{�}_1)$.

Step 2: Use the Slope (m):

• Use the given slope $�$.

Step 3: Use the Point-Slope Formula:

• Apply the point-slope formula with the chosen point $({�}_1,{�}_1)$ and the given slope $�$: $�-{�}_1=�(�-{�}_1)$

Step 4: Simplify the Equation:

• Simplify the equation to make it easier to work with.

Step 5: Convert to Standard Form:

• Rearrange the equation to standard form ($��+��=�$) by moving all terms to one side of the equation and ensuring that coefficients $�$, $�$, and $�$ are integers with no common factors other than 1.

Let's work through an example:

Example: Find the equation of the line passing through the points (2, 4) and (5, 10) and write it in standard form.

Step 1: Calculate the Slope (m):

• Calculate the slope $�$ using the two points: $�=\frac{10-4}{5-2}=\frac{6}{3}=2$

Step 2: Use One of the Points (x1, y1):

• Choose one of the points, say $(2,4)$.

Step 3: Use the Point-Slope Formula:

• Apply the point-slope formula with the calculated slope $�$ and the chosen point $(2,4)$: $�-4=2(�-2)$

Step 4: Simplify the Equation:

• Simplify the equation: $�-4=2�-4$

Step 5: Convert to Standard Form:

• Rearrange the equation to standard form: $�-2�=0$

So, the equation of the line passing through the points (2, 4) and (5, 10) in standard form is $�-2�=0$.

Vertical and horizontal lines are special cases of linear equations that have distinct characteristics. Here's what you need to know about them:

Vertical Line:

• A vertical line is a line that goes straight up and down. It is parallel to the y-axis in the Cartesian coordinate system.
• The equation of a vertical line can be written in the form $�=�$, where $�$ is the x-coordinate of any point on the line.
• For all points on a vertical line, the x-coordinate is the same, while the y-coordinate can vary.
• The slope of a vertical line is undefined because it does not have a change in x ($\mathrm{\Delta }�=0$).

Horizontal Line:

• A horizontal line is a line that goes straight left and right. It is parallel to the x-axis in the Cartesian coordinate system.
• The equation of a horizontal line can be written in the form $�=�$, where $�$ is the y-coordinate of any point on the line.
• For all points on a horizontal line, the y-coordinate is the same, while the x-coordinate can vary.
• The slope of a horizontal line is $0$ because it does not have a change in y ($\mathrm{\Delta }�=0$).

Here are some additional details:

• Vertical lines have undefined slopes because the change in x ($\mathrm{\Delta }�$) is zero, and division by zero is undefined in mathematics.
• Horizontal lines have slopes of $0$ because the change in y ($\mathrm{\Delta }�$) is zero, which means there is no rise in the line as you move along it.
• Vertical lines have equations in the form $�=�$ because they pass through all points with the same x-coordinate $�$.
• Horizontal lines have equations in the form $�=�$ because they pass through all points with the same y-coordinate $�$.

Here are some examples:

• The equation $�=3$ represents a vertical line passing through the point (3, 0).
• The equation $�=-2$ represents a horizontal line passing through the point (0, -2).
• The equation $�=0$ represents a vertical line passing through the origin (0, 0).
• The equation $�=5$ represents a horizontal line passing through the point (0, 5).

Vertical and horizontal lines have unique properties and are often used in geometry, graphing, and engineering applications due to their simplicity and predictability.

To find the equation of a line passing through two given points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$, you can follow these steps:

Step 1: Calculate the Slope ($�$):

• Calculate the slope ($�$) of the line using the two points: $�=\frac{{�}_2-{�}_1}{{�}_2-{�}_1}$

Step 2: Choose One of the Points (x1, y1):

• Select one of the two given points, say $({�}_1,{�}_1)$.

Step 3: Use the Point-Slope Formula:

• Apply the point-slope formula with the calculated slope $�$ and the chosen point $({�}_1,{�}_1)$: $�-{�}_1=�(�-{�}_1)$

Step 4: Simplify the Equation:

• Simplify the equation to make it easier to work with.

Here's an example to illustrate this process:

Example: Find the equation of the line passing through the points (2, 4) and (5, 10).

Step 1: Calculate the Slope ($�$):

• Calculate the slope $�$ using the two points: $�=\frac{10-4}{5-2}=\frac{6}{3}=2$

Step 2: Choose One of the Points (x1, y1):

• Let's choose the point (2, 4) as $({�}_1,{�}_1)$.

Step 3: Use the Point-Slope Formula:

• Apply the point-slope formula with the calculated slope $�$ and the chosen point (2, 4): $�-4=2(�-2)$

Step 4: Simplify the Equation:

• Simplify the equation: $�-4=2�-4$

Now, you have the equation of the line in slope-intercept form. If you want to express it in standard form ($��+��=�$), you can rearrange the equation:

$�-2�=0$

So, the equation of the line passing through the points (2, 4) and (5, 10) is $�-2�=0$.

To determine whether two lines are parallel or perpendicular based on their equations, you need to examine the slopes of the lines. Here are the rules for determining the relationship between two lines:

1. Parallel Lines:

   • Two lines are parallel if and only if their slopes are equal.
   • If the slopes of two lines are the same, then the lines are parallel.
   • The equations of two parallel lines will have the same slope.

2. Perpendicular Lines:

   • Two lines are perpendicular if and only if the product of their slopes is equal to -1.
   • If the slopes of two lines, say ${�}_1$ and ${�}_2$, satisfy the equation ${�}_1\cdot {�}_2=-1$, then the lines are perpendicular.
   • In other words, the slope of one line is the negative reciprocal of the slope of the other line.
   • The negative reciprocal of a slope $�$ is $-1\mathrm{/}�$.

Let's look at examples:

Example 1: Parallel Lines Consider the equations of two lines:

1. Line 1: $�=2�+3$
2. Line 2: $�=2�-1$

Both lines have the same slope, which is 2. Therefore, Line 1 and Line 2 are parallel.

Example 2: Perpendicular Lines Consider the equations of two lines:

1. Line 1: $�=3�+2$
2. Line 2: $�=-1\mathrm{/}3�+5$

To check if they are perpendicular, calculate their slopes. The slope of Line 1 is 3, and the slope of Line 2 is $-1\mathrm{/}3$. Now, check if their product is -1:

$3\cdot (-1\mathrm{/}3)=-1$

The product of their slopes is -1, which means Line 1 and Line 2 are perpendicular.

In summary, you can determine whether two lines are parallel or perpendicular by comparing the slopes of their equations. If the slopes are equal, the lines are parallel, and if the product of their slopes is -1, the lines are perpendicular.

To determine whether two lines are parallel, perpendicular, or neither based on their equations, you'll first need to graph the lines and then examine their slopes. Here are the steps:

Step 1: Graph the Two Lines:

• Graph both equations on the same coordinate system. You can do this by plotting a few points for each line and connecting them to form the lines.

Step 2: Calculate the Slopes of the Lines:

• Once you have the graphs, calculate the slopes (${�}_1$ and ${�}_2$) of the two lines.

Step 3: Determine the Relationship:

• Based on the slopes, determine the relationship between the lines:
  • If ${�}_1={�}_2$, the lines are parallel.
  • If ${�}_1\cdot {�}_2=-1$, the lines are perpendicular.
  • If ${�}_1$ and ${�}_2$ have different values and their product is not -1, the lines are neither parallel nor perpendicular.

Let's work through an example:

Example: Determine whether the lines represented by the equations $�=2�+3$ and $�=-1\mathrm{/}2�-1$ are parallel, perpendicular, or neither.

Step 1: Graph the Two Lines:

• Plot points for both lines and draw the graphs on the same coordinate system.

Step 2: Calculate the Slopes of the Lines:

• For the first line ($�=2�+3$), the slope (${�}_1$) is 2.
• For the second line ($�=-1\mathrm{/}2�-1$), the slope (${�}_2$) is -1/2.

Step 3: Determine the Relationship:

• Compare the slopes:
  • ${�}_1=2$ and ${�}_2=-1\mathrm{/}2$.
  • Since ${�}_1\cdot {�}_2=2\cdot (-1\mathrm{/}2)=-1$, the lines have slopes that multiply to -1, which means they are perpendicular.

So, in this example, the lines represented by the equations $�=2�+3$ and $�=-1\mathrm{/}2�-1$ are perpendicular.

To write the equations of lines that are parallel or perpendicular to a given line, you'll need to know the equation of the given line and the desired relationship (parallel or perpendicular). Here's how to do it:

Given Line Equation: Let's say you have the equation of a given line in slope-intercept form:

$�=��+�$

Where:

• $�$ is the slope of the given line.
• $�$ is the y-intercept of the given line.

Parallel Line:

• To find the equation of a line that is parallel to the given line, you'll want to keep the same slope $�$ but possibly change the y-intercept ($�$).

The equation of a parallel line will also be in the form $�=��+{�}_1$, where ${�}_1$ is the new y-intercept you choose.

Perpendicular Line:

• To find the equation of a line that is perpendicular to the given line, you'll need to use the negative reciprocal of the slope $�$ of the given line.

The equation of a perpendicular line will be in the form $�=-1\mathrm{/}��+{�}_2$, where ${�}_2$ is the new y-intercept you choose.

Here are the steps:

For a Parallel Line:

1. Keep the slope $�$ the same.
2. Choose a new y-intercept (${�}_1$) for the parallel line.
3. Write the equation in the form $�=��+{�}_1$.

For a Perpendicular Line:

1. Calculate the negative reciprocal of the slope $�$ of the given line. The negative reciprocal is $-1\mathrm{/}�$.
2. Choose a new y-intercept (${�}_2$) for the perpendicular line.
3. Write the equation in the form $�=-1\mathrm{/}��+{�}_2$.

Let's work through an example for each case:

Example 1: Finding the Equation of a Line Parallel to a Given Line:

Given Line: $�=2�+3$

To find a line parallel to this line, we keep the slope (2) the same and choose a new y-intercept. Let's say we want the new line to pass through the point (1, 4).

New Parallel Line: $�=2�+2$ (slope remains 2, and the y-intercept is changed to 2).

Example 2: Finding the Equation of a Line Perpendicular to a Given Line:

Given Line: $�=-1\mathrm{/}3�+5$

To find a line perpendicular to this line, we calculate the negative reciprocal of the slope (-1/3), which is 3. Let's say we want the new line to pass through the point (2, 4).

New Perpendicular Line: $�=3�+2$ (slope is the negative reciprocal of -1/3, and the y-intercept is 2).

These are examples of how to find the equations of lines parallel or perpendicular to a given line by manipulating the slope and, if needed, the y-intercept.

Let's work through a couple of examples to find the equations of lines that are parallel or perpendicular to given lines.

Example 1: Finding the Equation of a Line Parallel to a Given Line

Given Line: $�=2�-1$

We want to find the equation of a line that is parallel to the given line and passes through the point (3, 5).

Solution:

1. The slope of the given line is $�=2$.
2. To find a line parallel to it, we keep the slope the same.
3. Now, we have the slope ($�$) and a point (3, 5). Use the point-slope form to write the equation: $�-5=2(�-3)$
4. Simplify the equation: $�-5=2�-6$
5. Isolate $�$ on the left side: $�=2�-6+5$ $�=2�-1$

So, the equation of the line parallel to $�=2�-1$ that passes through (3, 5) is $�=2�-1$.

Example 2: Finding the Equation of a Line Perpendicular to a Given Line

Given Line: $�=-3�+4$

We want to find the equation of a line that is perpendicular to the given line and passes through the point (2, -1).

Solution:

1. The slope of the given line is $�=-3$.
2. To find a line perpendicular to it, we calculate the negative reciprocal of the slope: $-1\mathrm{/}(-3)=1\mathrm{/}3$.
3. Now, we have the new slope ($�$) and a point (2, -1). Use the point-slope form to write the equation: $�-(-1)=(1\mathrm{/}3)(�-2)$
4. Simplify the equation: $�+1=(1\mathrm{/}3)(�-2)$
5. Isolate $�$ on the left side: $�=(1\mathrm{/}3)(�-2)-1$ $�=(1\mathrm{/}3)�-2\mathrm{/}3-1$ $�=(1\mathrm{/}3)�-2\mathrm{/}3-3\mathrm{/}3$ $�=(1\mathrm{/}3)�-5\mathrm{/}3$

So, the equation of the line perpendicular to $�=-3�+4$ that passes through (2, -1) is $�=(1\mathrm{/}3)�-5\mathrm{/}3$.

These examples illustrate how to find equations of lines parallel or perpendicular to given lines using the slope and point-slope form.

To find the equation of a line that is perpendicular to a given line and passes through a specific point, you'll need to follow these steps:

Given Line Equation: Let's say you have the equation of a given line in slope-intercept form:

$�=��+�$

Where:

• $�$ is the slope of the given line.
• $�$ is the y-intercept of the given line.

Perpendicular Line:

• To find the equation of a line that is perpendicular to the given line, you'll need to use the negative reciprocal of the slope $�$ of the given line.

The equation of a perpendicular line will be in the form $�=-1\mathrm{/}��+{�}_1$, where ${�}_1$ is the new y-intercept you choose based on the point through which the line passes.

Here are the steps:

For a Perpendicular Line:

1. Calculate the negative reciprocal of the slope $�$ of the given line. The negative reciprocal is $-1\mathrm{/}�$.
2. Choose the point $({�}_1,{�}_1)$ through which the perpendicular line should pass.
3. Use the calculated slope $-1\mathrm{/}�$ and the chosen point $({�}_1,{�}_1)$ to write the equation in the form $�=-1\mathrm{/}��+{�}_1$.
4. Simplify the equation by solving for ${�}_1$.

Let's work through an example:

Example: Find the equation of a line that is perpendicular to the given line $�=2�-1$ and passes through the point (3, 5).

Step 1: Calculate the Negative Reciprocal of the Slope:

• The slope of the given line is $�=2$.
• The negative reciprocal of 2 is $-1\mathrm{/}2$.

Step 2: Choose the Point:

• We want the perpendicular line to pass through the point (3, 5).

Step 3: Use the Point and Slope:

• Use the negative reciprocal slope $-1\mathrm{/}2$ and the chosen point (3, 5) to write the equation: $�-5=(-1\mathrm{/}2)(�-3)$

Step 4: Simplify the Equation:

• Simplify the equation: $�-5=(-1\mathrm{/}2)�+3\mathrm{/}2$

Now, isolate $�$ on the left side of the equation:

$�=(-1\mathrm{/}2)�+3\mathrm{/}2+5$

$�=(-1\mathrm{/}2)�+3\mathrm{/}2+10\mathrm{/}2$

$�=(-1\mathrm{/}2)�+13\mathrm{/}2$

So, the equation of the line that is perpendicular to $�=2�-1$ and passes through the point (3, 5) is $�=(-1\mathrm{/}2)�+13\mathrm{/}2$.

To solve the equation $3(2�-1)+�=5�+3$, follow these steps:

Step 1: Distribute the 3 on the left side: $6�-3+�=5�+3$

Step 2: Combine like terms on both sides: $7�-3=5�+3$

Step 3: Isolate the variable $�$ on one side of the equation: Subtract $5�$ from both sides:

$7�-5�-3=3$

Simplify:

$2�-3=3$

Step 4: Add 3 to both sides to isolate $2�$: $2�-3+3=3+3$

Simplify:

$2�=6$

Step 5: Solve for $�$ by dividing both sides by 2: $2�\mathrm{/}2=6\mathrm{/}2$

Simplify:

$�=3$

So, the solution to the equation $3(2�-1)+�=5�+3$ is $�=3$.