Question

solve the following system of equations graphically on the set of axes below.
$y = x + 2$
$y = -dfrac{1}{5}x + 8$
plot two lines by clicking the graph.
click a line to delete it.
graph with x-axis from -10 to 10 and y-axis from -10 to 10, grid lines, and axes arrows

Explanation:

Step1: Graph \( y = x + 2 \)

The equation \( y = x + 2 \) is in slope - intercept form (\( y=mx + b \)) where the slope \( m = 1 \) and the y - intercept \( b = 2 \). To graph this line, we start by plotting the y - intercept at the point \( (0,2) \). Then, using the slope (rise over run), since the slope is 1 (which can be written as \( \frac{1}{1} \)), we move 1 unit up and 1 unit to the right from the point \( (0,2) \) to get the point \( (1,3) \), and we can continue this process to plot more points or draw the line.

Step2: Graph \( y=-\frac{1}{5}x + 8 \)

The equation \( y =-\frac{1}{5}x + 8 \) is also in slope - intercept form with slope \( m=-\frac{1}{5} \) and y - intercept \( b = 8 \). We plot the y - intercept at the point \( (0,8) \). The slope \( -\frac{1}{5} \) means we move 1 unit down and 5 units to the right (or 1 unit up and 5 units to the left) from the y - intercept. So from \( (0,8) \), moving 5 units to the right and 1 unit down gives us the point \( (5,7) \), and we can use this to draw the line.

Step3: Find the intersection point

The solution to the system of equations is the point where the two lines intersect. To find this point algebraically (to confirm the graphical solution), we set the two equations equal to each other:

$$
LATEXBLOCK0
$$

Now substitute \( x = 5 \) into the first equation \( y=x + 2 \), we get \( y=5 + 2=7 \). So the intersection point (the solution to the system) is \( (5,7) \). When we graph the two lines, they will intersect at the point \( (5,7) \).

Answer:

The solution to the system of equations is \( (5,7) \)