Problem 18

Question

Find the equation of the line that contains the two given points. Express equations in the form $A x+B y=C$, where $A, B$, and $C$ are integers. (Objective $1 \mathrm{~b}$ ) $(-8,-7)$ and $(-3,-1)$

Step-by-Step Solution

Verified

Answer

The equation of the line is $6x - 5y = -13$.

1Step 1: Identify the Slope

First, we need to find the slope (m) of the line that passes through the two points $(-8, -7)$ and $(-3, -1)$. The formula for the slope is: m = \frac{y_2 - y_1}{x_2 - x_1}. Substituting the given points, we have: m = \frac{-1 - (-7)}{-3 - (-8)} = \frac{-1 + 7}{-3 + 8} = \frac{6}{5}.

2Step 2: Use the Point-Slope Form

Now that we have the slope $m = \frac{6}{5}$, we use the point-slope form equation, which is: y - y_1 = m(x - x_1)\. Using the point $(-8, -7)$, the equation becomes: y + 7 = \frac{6}{5}(x + 8).

3Step 3: Simplify to Slope-Intercept Form

Distribute the slope on the right side: $y + 7 = \frac{6}{5}x + \frac{6}{5} \cdot 8$ results in y + 7 = \frac{6}{5}x + \frac{48}{5}. Subtract 7 from both sides to isolate y: $y = \frac{6}{5}x + \frac{48}{5} - 7$ Convert $7$ into fifths: $y = \frac{6}{5}x + \frac{48}{5} - \frac{35}{5}$, resulting in $y = \frac{6}{5}x + \frac{13}{5}$.

4Step 4: Convert to Standard Form

To express the equation in the form $Ax + By = C$, first eliminate the fraction by multiplying the entire equation by 5: $5y = 6x + 13$. Subtract $6x$ from both sides to bring $x$ to the left side: $-6x + 5y = 13$. Finally, multiply through by -1 to keep the leading coefficient positive: $6x - 5y = -13$.

Key Concepts

Slope calculationPoint-slope formSlope-intercept formStandard form conversion

Slope calculation

To find the equation of a line, one of the first steps is calculating the slope, represented by the symbol $ m $. The slope tells us how steep the line is and the direction it goes. To calculate the slope between two points, we use the formula:

$ m = \frac{y_2 - y_1}{x_2 - x_1} $

where $(x_1, y_1)$ and $(x_2, y_2)$ are the given points.
In this exercise, our points are $(-8, -7)$ and $(-3, -1)$.
Substituting these values into the formula, we find:

$ m = \frac{-1 - (-7)}{-3 - (-8)} = \frac{6}{5} $

This means the line rises 6 units for every 5 units it moves to the right.

Point-slope form

Once we have the slope, the next step is to use the point-slope form. This form is useful to find the equation of a line precisely when you know one point on the line and the slope.
The point-slope form equation is given by:

$ y - y_1 = m(x - x_1) $

Using the slope $ m = \frac{6}{5} $ and one of the points, let's choose $(-8, -7)$,
the equation becomes:

$ y + 7 = \frac{6}{5}(x + 8) $

Articulating "+" is because the point $(-8, -7)$ involves subtracting the negatives.
This form makes it easy to substitute and see how changes in $ x $ affect $ y $.

Slope-intercept form

The slope-intercept form of a line equation is easy to understand and widely used.
It is of the form $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept.
To convert our point-slope form equation to this, we distribute and simplify:

$ y + 7 = \frac{6}{5}x + \frac{48}{5} $

Subtract 7 from the sides to isolate $ y $:

$ y = \frac{6}{5}x + \frac{48}{5} - 7 $

Change 7 into a fraction with a 5 denominator:

$ 7 = \frac{35}{5} $

Now it becomes:

$ y = \frac{6}{5}x + \frac{13}{5} $

This shows the line's slope is $ \frac{6}{5} $, intersecting the y-axis at $ \frac{13}{5} $.

Standard form conversion

To complete the equation transformation, we change the slope-intercept form into standard form.
The standard form is expressed as $ Ax + By = C $, with $ A $, $ B $, and $ C $ being integers.
Starting from the slope-intercept form: