Question

graph the system of equations.
$y = \frac{4}{5}x + 1$
$y = -\frac{1}{5}x + 6$

Explanation:

Step1: Analyze the first equation \( y = \frac{4}{5}x + 1 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m=\frac{4}{5} \) and the y - intercept \( b = 1 \). To graph this line, we can start by plotting the y - intercept. The y - intercept is the point where \( x = 0 \), so when \( x=0 \), \( y=1 \). Then, using the slope, from the point \( (0,1) \), we can move up 4 units and to the right 5 units (since slope \(=\frac{\text{rise}}{\text{run}}=\frac{4}{5}\)) to get another point on the line, for example, when \( x = 5 \), \( y=\frac{4}{5}(5)+1=4 + 1=5 \), so the point \( (5,5) \) is also on this line.

Step2: Analyze the second equation \( y=-\frac{1}{5}x + 6 \)

This is also a linear equation in slope - intercept form with slope \( m=-\frac{1}{5} \) and y - intercept \( b = 6 \). Plot the y - intercept first: when \( x = 0 \), \( y = 6 \). Then, using the slope, from the point \( (0,6) \), we can move down 1 unit and to the right 5 units (since slope \(=\frac{\text{rise}}{\text{run}}=\frac{- 1}{5}\)) to get another point. For example, when \( x = 5 \), \( y=-\frac{1}{5}(5)+6=-1 + 6 = 5 \), so the point \( (5,5) \) is also on this line.

Step3: Find the intersection point

To find the solution of the system of equations (the point where the two lines intersect), we can set the two equations equal to each other:
\[

$$\begin{align*} \frac{4}{5}x+1&=-\frac{1}{5}x + 6\\ \frac{4}{5}x+\frac{1}{5}x&=6 - 1\\ \frac{4x + x}{5}&=5\\ \frac{5x}{5}&=5\\ x&=5 \end{align*}$$

\]
Substitute \( x = 5 \) into the first equation \( y=\frac{4}{5}(5)+1=4 + 1=5 \). So the solution of the system is \( (5,5) \). When graphing, the two lines should intersect at the point \( (5,5) \). The first line has a positive slope (\( \frac{4}{5} \)) and passes through \( (0,1) \) and \( (5,5) \), the second line has a negative slope (\( -\frac{1}{5} \)) and passes through \( (0,6) \) and \( (5,5) \).

Answer:

The system of equations \( y=\frac{4}{5}x + 1 \) and \( y=-\frac{1}{5}x+6 \) intersects at the point \( (5,5) \). To graph the system, plot the y - intercepts \( (0,1) \) for the first line and \( (0,6) \) for the second line, then use the slopes to find additional points on each line. The two lines intersect at \( (5,5) \).