Question

here is a system of equations.
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$$\begin{cases} y = -2x + 1 \\\\ y = \\frac{1}{4}x - 2 \\end{cases}$$

(a) graph the system.
(b) use your graph to estimate the solution to the system. write the coordinates to the tenths place. (you will be given some leeway in your answer.)

Explanation:

Step1: Find intercepts for $y=-2x+1$

For x-intercept: set $y=0$
$0=-2x+1 \implies x=\frac{1}{2}=0.5$, so point $(0.5, 0)$
For y-intercept: set $x=0$
$y=-2(0)+1=1$, so point $(0, 1)$

Step2: Find intercepts for $y=\frac{1}{4}x-2$

For x-intercept: set $y=0$
$0=\frac{1}{4}x-2 \implies \frac{1}{4}x=2 \implies x=8$ (outside graph, use another point: set $x=4$
$y=\frac{1}{4}(4)-2=1-2=-1$, so point $(4, -1)$
For y-intercept: set $x=0$
$y=\frac{1}{4}(0)-2=-2$, so point $(0, -2)$

Step3: Graph both lines

Plot points for $y=-2x+1$: $(0,1)$ and $(0.5,0)$, draw line.
Plot points for $y=\frac{1}{4}x-2$: $(0,-2)$ and $(4,-1)$, draw line.

Step4: Find intersection (solve algebraically for exact value, then estimate)

Set $-2x+1=\frac{1}{4}x-2$
Multiply all terms by 4: $-8x+4=x-8$
Rearrange: $-8x-x=-8-4 \implies -9x=-12 \implies x=\frac{12}{9}=\frac{4}{3}\approx1.3$
Substitute $x=\frac{4}{3}$ into $y=-2x+1$:
$y=-2(\frac{4}{3})+1=-\frac{8}{3}+\frac{3}{3}=-\frac{5}{3}\approx-1.7$

Answer:

(a) (Graph: Draw line 1 through $(0,1)$ and $(0.5,0)$; Draw line 2 through $(0,-2)$ and $(4,-1)$)
(b) $(1.3, -1.7)$