Question

On a coordinate plane, a line goes through points (0, 2) and (2, 0).
Caleb graphed the linear function y = –x + 2 of a system of equations. The other linear function passes through the points (–2, –8) and (1, –2).

What is the slope of the linear function that passes through the points (–2, –8) and (1, –2)?

What is the y-intercept?

Graph the linear function to determine the solution. What is the solution?

Answer 1

To find the slope and y-intercept of the linear function passing through the points \((-2, -8)\) and \((1, -2)\), we can follow these steps:

### Step 1: Calculate the Slope
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Plugging in the given points \((-2, -8)\) as \((x_1, y_1)\) and \((1, -2)\) as \((x_2, y_2)\):

\[
m = \frac{-2 - (-8)}{1 - (-2)} = \frac{-2 + 8}{1 + 2} = \frac{6}{3} = 2
\]

So, the slope of the linear function is \(2\).

### Step 2: Find the Y-Intercept
To find the y-intercept, we can use the slope-intercept form of a line, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We can substitute one of the points and the slope into this equation to solve for \(b\).

Using the point \((-2, -8)\):

\[
-8 = 2(-2) + b
\]
\[
-8 = -4 + b
\]
\[
b = -8 + 4 = -4
\]

Thus, the y-intercept is \(-4\).

### Step 3: Write the Equation of the Line
Using the slope \(2\) and the y-intercept \(-4\), the equation of the line can be written as:

\[
y = 2x - 4
\]

### Step 4: Graph the Functions
1. The first line (Caleb's line) is \(y = -x + 2\).
2. The second line is \(y = 2x - 4\).

### Step 5: Find the Intersection (Solution)
To find the solution of the system, we set the two equations equal to each other:

\[
-x + 2 = 2x - 4
\]

Now, solve for \(x\):

\[
2 + 4 = 2x + x
\]
\[
6 = 3x
\]
\[
x = 2
\]

Now, substitute \(x = 2\) back into either equation to solve for \(y\). Using Caleb’s line:

\[
y = -2 + 2 = 0
\]

Thus, the solution to the system of equations is the point \((2, 0)\).

### Summary:
- **Slope of the second line**: \(2\)
- **Y-intercept of the second line**: \(-4\)
- **Solution to the system of equations**: \((2, 0)\)