Question

1. solve graphically: developing

$y = x - 4$

$y = -\dfrac{2}{3}x + 1$

Explanation:

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

x-intercept: set $y=0$, $0=x-4 \implies x=4$ (point $(4,0)$)
y-intercept: set $x=0$, $y=0-4=-4$ (point $(0,-4)$)

Step2: Find intercepts for $y=-\frac{2}{3}x+1$

x-intercept: set $y=0$, $0=-\frac{2}{3}x+1 \implies x=\frac{3}{2}=1.5$ (point $(1.5,0)$)
y-intercept: set $x=0$, $y=0+1=1$ (point $(0,1)$)

Step3: Plot lines and find intersection

Plot both lines using their intercepts. The lines intersect at $(3, -1)$.
Verify algebraically:
Set $x-4 = -\frac{2}{3}x+1$
$x+\frac{2}{3}x = 1+4$
$\frac{5}{3}x = 5$
$x=3$
Substitute $x=3$ into $y=x-4$: $y=3-4=-1$

Answer:

The solution is the point $(3, -1)$