Question

practice
solve each system of equations by graphing.

1. $y = x + 4$

$y = -2x - 2$
(grid with x and y axes shown)

Explanation:

Step1: Analyze the first equation \( y = x + 4 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m = 1 \) and the y - intercept \( b = 4 \). To graph this line, we can start by plotting the y - intercept at the point \( (0,4) \). Then, using the slope (rise over run, since \( m = 1=\frac{1}{1} \)), we can find another point. From \( (0,4) \), we move 1 unit up and 1 unit to the right to get the point \( (1,5) \), or 1 unit down and 1 unit to the left to get the point \( (- 1,3) \).

Step2: Analyze the second equation \( y=-2x - 2 \)

This is also a linear equation in slope - intercept form with slope \( m=-2 \) and y - intercept \( b = - 2 \). We plot the y - intercept at the point \( (0,-2) \). Using the slope \( m=-2=\frac{-2}{1} \), from the point \( (0,-2) \), we move 2 units down and 1 unit to the right to get the point \( (1,-4) \), or 2 units up and 1 unit to the left to get the point \( (-1,0) \).

Step3: Find the intersection point

When we graph both lines, we look for the point where they cross each other. To find the intersection algebraically (which can be used to verify the graph), we set the two equations equal to each other since at the intersection point, the y - values (and x - values) are equal.
Set \( x + 4=-2x-2 \)
Add \( 2x \) to both sides of the equation: \( x+2x + 4=-2x + 2x-2 \), which simplifies to \( 3x+4=-2 \)
Subtract 4 from both sides: \( 3x+4 - 4=-2 - 4 \), so \( 3x=-6 \)
Divide both sides by 3: \( x=\frac{-6}{3}=-2 \)
Now substitute \( x = - 2 \) into the first equation \( y=x + 4 \). Then \( y=-2 + 4=2 \)

Answer:

The solution to the system of equations \(

$$\begin{cases}y=x + 4\\y=-2x-2\end{cases}$$

\) is the point of intersection of the two lines, which is \( (-2,2) \)