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.

Explanation:

Step1: Find intercepts for $y=-x+4$

When $x=0$, $y=4$; when $y=0$, $x=4$. So points are $(0,4)$ and $(4,0)$.

Step2: Rewrite second equation to slope-intercept

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

Step3: Find intercepts for $y=\frac{5}{2}x-3$

When $x=0$, $y=-3$; when $y=0$, $0=\frac{5}{2}x-3 \implies x=\frac{6}{5}=1.2$. So points are $(0,-3)$ and $(1.2,0)$.

Step4: Find intersection algebraically (verify)

Substitute $y=-x+4$ into $5x-2y=6$:

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

Substitute $x=2$ into $y=-x+4$: $y=-2+4=2$.

Answer:

The solution to the system is $(2, 2)$ (the intersection point of the two lines plotted using their intercepts: $y=-x+4$ uses $(0,4)$ and $(4,0)$; $y=\frac{5}{2}x-3$ uses $(0,-3)$ and $(1.2,0)$).