Problem 15 - Algebra A Combined Function

Question

Write an equation of the line passing through the given points. Write the equation in standard form \(A x+B y=C\). See Example 2 . (-3,-8) and (-6,-9)

Step-by-Step Solution

Answer

The equation of the line is \( x - 3y = 21 \).

Step 1: Find the Slope

First, we will find the slope of the line passing through the points (-3,-8) and (-6,-9). The formula to calculate the slope \( m \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the values, we get \( m = \frac{-9 - (-8)}{-6 - (-3)} = \frac{-9 + 8}{-6 + 3} = \frac{-1}{-3} = \frac{1}{3} \). So, the slope is \( \frac{1}{3} \).

Step 2: Use Point-Slope Form

Now that we know the slope, we use the point-slope form of a line equation, which is \( y - y_1 = m(x - x_1) \). We can use one of the points, say (-3,-8). Substitute \( m = \frac{1}{3} \), \( x_1 = -3 \), and \( y_1 = -8 \) into the formula: \( y + 8 = \frac{1}{3}(x + 3) \).

Step 3: Simplify to Slope-Intercept Form

Distribute the \( \frac{1}{3} \) on the right side: \( y + 8 = \frac{1}{3}x + 1 \). Subtract 8 from both sides to isolate \( y \): \( y = \frac{1}{3}x + 1 - 8 \). This simplifies to \( y = \frac{1}{3}x - 7 \).

Step 4: Convert to Standard Form

We need to express the equation \( y = \frac{1}{3}x - 7 \) in the standard form \( Ax + By = C \). Start by eliminating the fraction. Multiply the entire equation by 3 to get: \( 3y = x - 21 \). Rearrange it to have \( x \) and \( y \) on one side: \( -x + 3y = -21 \).

Step 5: Adjust to Positive A

The standard form of the equation prefers \( A \) to be non-negative. Multiply through by -1 to achieve this: \( x - 3y = 21 \). Now this is the equation in the standard form \( Ax + By = C \).

Key Concepts

Slope Calculation

The first step in finding the equation of a line is calculating its slope. The slope tells us how steep the line is. We can find it using two points of the line. The formula to calculate the slope, often denoted as \( m \), is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
In the given problem, two points are (-3,-8) and (-6,-9). Plug these values into the formula:

• For \( y_2 - y_1 \), calculate \(-9 - (-8)\) which simplifies to \(-9 + 8 = -1\).
• For \( x_2 - x_1 \), calculate \(-6 - (-3)\) which simplifies to \(-6 + 3 = -3\).
• This gives us a slope \( m = \frac{-1}{-3} = \frac{1}{3} \).

The slope of \( \frac{1}{3} \) means that for every unit increase in \( x \), \( y \) increases by \( \frac{1}{3} \). This is crucial for understanding the line's direction and steepness.

Point-Slope Form

Once we have the slope, the next step is to use the point-slope form of a line to find its equation. The point-slope form is represented by the formula \( y - y_1 = m(x - x_1) \).
We choose one of our known points, such as (-3,-8), and use it along with our calculated slope \( \frac{1}{3} \). Substituting the values, we get:

• \( y + 8 = \frac{1}{3}(x + 3) \)

This form is very helpful because it directly uses a known point and the slope, making it easy to switch into other forms of the line equation later. The point-slope form is especially beneficial when working with different points since it gives immediate insights into how changes between points affect the line.

Slope-Intercept Form

The slope-intercept form of a line is a popular approach that directly shows the slope and the y-intercept of the line. It is expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
From our point-slope equation \( y + 8 = \frac{1}{3}(x + 3) \), we can transform it into slope-intercept form. Distribute \( \frac{1}{3} \):

• \( y + 8 = \frac{1}{3}x + 1 \)
• Subtract 8 from both sides to solve for \( y \): \( y = \frac{1}{3}x + 1 - 8 \)
• This simplifies to \( y = \frac{1}{3}x - 7 \)

The equation \( y = \frac{1}{3}x - 7 \) clearly indicates that the y-intercept is -7. This form allows you to quickly graph the line or understand how changes in \( x \) affect \( y \).

Equation Transformation

Finally, we need to express the equation in standard form, which is \( Ax + By = C \). The process involves rearranging and transforming the slope-intercept form to meet these requirements.
Our slope-intercept equation is \( y = \frac{1}{3}x - 7 \). To eliminate the fraction, multiply every term by 3:

• \( 3y = x - 21 \)

Next, rearrange to get all terms on one side:

• \( -x + 3y = -21 \)

To ensure \( A \) is positive, multiply the entire equation by -1:

• \( x - 3y = 21 \)

This gives us the standard form \( x - 3y = 21 \). It's a universally recognized format for line equations and is particularly useful for solving systems of equations, as it aligns nicely with algebraic solving methods.