Question

solve the following system of equations graphically on the set of axes below.
$y = -x + 4$
$5x - 2y = 6$
plot two lines by clicking the graph.
click a line to delete it.
answer attempt 1 out of 2
solution: 1 submit answer

Explanation:

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

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

Step2: Rewrite the second equation \( 5x-2y = 6 \) in slope - intercept form

We solve for \( y \):
\[

$$\begin{align*} 5x-2y&=6\\ -2y&=-5x + 6\\ y&=\frac{5}{2}x-3 \end{align*}$$

\]
This is also a linear equation in slope - intercept form with slope \( m = \frac{5}{2} \) and y - intercept \( b=-3 \). To graph this line, we start by plotting the y - intercept at \( (0,-3) \). Then, using the slope \( \frac{5}{2} \) (rise 5 units, run 2 units), from the point \( (0,-3) \), we can move 5 units up and 2 units to the right to get the point \( (2,2) \), or 5 units down and 2 units to the left to get \( (-2,-8) \).

Step3: Find the intersection point of the two lines

To find the solution of the system of equations, we can also solve the system algebraically (to verify the graphical solution). We have the system:
\[

$$\begin{cases} y=-x + 4\\ 5x-2y=6 \end{cases}$$

\]
Substitute \( y=-x + 4 \) into the second equation:
\[

$$\begin{align*} 5x-2(-x + 4)&=6\\ 5x + 2x-8&=6\\ 7x&=6 + 8\\ 7x&=14\\ x&=2 \end{align*}$$

\]
Then substitute \( x = 2 \) into \( y=-x + 4 \):
\( y=-2 + 4=2 \)

So the solution to the system of equations (the point of intersection of the two lines) is \( (2,2) \).

Answer:

The solution to the system of equations is \( (2,2) \)