Problem 41 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the points. $$ (-8,9),(10,-3) $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the points (-8,9) and (10,-3) is \( y = -2/3x + 5 \).

Step 1: Identify the Given Points

The points given in this problem are (-8,9) and (10,-3). Thus, we can identify x1 = -8, y1 = 9, x2 = 10, and y2 = -3.

Step 2: Calculate the Slope

Next, we can calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Substituting the known values, we get m = (-3 - 9) / (10 - (-8)) = -12 / 18 = -2/3.

Step 3: Calculate the y-intercept

Using the equation y - y1 = m(x - x1), we substitute the values of the slope and one of the points, say (-8,9), into the equation to calculate the y-intercept. This gives us y - 9 = -2/3(x - (-8)). Simplifying, we find y = -2/3x + 5.

Key Concepts

Slope Calculation

To find the slope of a line passing through two points, use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula determines the rate of change between the two points in the coordinate plane, describing how much the y-value changes for every change in the x-value. In simpler terms, the slope tells us how steep the line is.
Imagine we are given two points: \((-8,9)\) and \((10,-3)\). With these coordinates, we can identify \(x_1 = -8\), \(y_1 = 9\), \(x_2 = 10\), and \(y_2 = -3\).
Substitute these values into the slope formula to calculate it:

• \((y_2 - y_1) = -3 - 9 = -12\)
• \((x_2 - x_1) = 10 - (-8) = 10 + 8 = 18\)

Therefore, the slope \(m = \frac{-12}{18} = -\frac{2}{3}\). This means that for every 3 units you move horizontally (to the right), the line moves 2 units downward. It's like a gentle downhill slope.

Y-Intercept Calculation

Once the slope is known, the next step is finding the y-intercept, which is the point where the line crosses the y-axis.
We use the point-slope form to do this: \[ y - y_1 = m(x - x_1) \] This form rearranges the points and slope into a convenient equation format.
For our example, we take the slope \(m = -\frac{2}{3}\) and one of the given points, say \((-8,9)\). Substituting these into the equation provides:

• Start with: \(y - 9 = -\frac{2}{3}(x - (-8))\)
• This simplifies to: \(y - 9 = -\frac{2}{3}(x + 8)\)
• Distribute and solve: \(y = -\frac{2}{3}x - \frac{16}{3} + 9\)

Converting into a single fraction, the y-intercept \(b\) is found to be 5. Thus, the line crosses the y-axis at \(y = 5\).

Equation of a Line

With the slope and y-intercept calculated, writing the equation of the line in slope-intercept form is straightforward.
The slope-intercept form of the equation of a line is written as: \[ y = mx + b \] Here, \(m\) represents the slope, and \(b\) stands for the y-intercept.
For our problem, we calculated the slope to be \(-\frac{2}{3}\) and the y-intercept to be 5. Plug these values into the slope-intercept formula:

• Replace \(m\) with \(-\frac{2}{3}\)
• Replace \(b\) with 5

Thus, the equation of the line that passes through the points \((-8,9)\) and \((10,-3)\) in slope-intercept form is: \[ y = -\frac{2}{3}x + 5 \] This equation succinctly describes how to calculate the y-value based on any x-value along this line.