Question

solve the following system of equations graphically on the set of axes below.
$y = -x + 5$
$x - 2y = 8$
plot two lines by clicking the graph.
click a line to delete it.
answer
solution: 1

Explanation:

Step1: Analyze the first equation \( y = -x + 5 \)

To graph this line, we can find two points. When \( x = 0 \), \( y = 5 \) (so the y - intercept is \( (0,5) \)). When \( y = 0 \), \( 0=-x + 5\), which gives \( x = 5 \) (so the x - intercept is \( (5,0) \)).

Step2: Analyze the second equation \( x-2y = 8 \)

We can rewrite it in slope - intercept form (\( y=mx + b \)):
\[

$$\begin{align*} x-2y&=8\\ -2y&=-x + 8\\ y&=\frac{1}{2}x-4 \end{align*}$$

\]
To find points, when \( x = 0 \), \( y=-4 \) (y - intercept \( (0, - 4) \)). When \( y = 0 \), \( 0=\frac{1}{2}x-4\), so \( \frac{1}{2}x=4 \) and \( x = 8 \) (x - intercept \( (8,0) \)).

Step3: Find the intersection point (solution)

We can also solve the system algebraically to find the exact solution (which will be the intersection point of the two lines).
Substitute \( y=-x + 5 \) into \( x-2y = 8 \):
\[

$$\begin{align*} x-2(-x + 5)&=8\\ x + 2x-10&=8\\ 3x&=8 + 10\\ 3x&=18\\ x&=6 \end{align*}$$

\]
Then substitute \( x = 6 \) into \( y=-x + 5 \), we get \( y=-6 + 5=-1 \).

Answer:

The solution to the system of equations (the point of intersection of the two lines) is \( (6,-1) \)