Question

question 1 (multiple choice worth 6 points) (06.02r mc)
choose the system of equations that matches the following graph:

1. $2x + 4y = 0$

$7x - 6y = -24$

1. $2x - 4y = 0$

$7x + 6y = -24$

1. $2x - 4y = 0$

$7x - 6y = -24$

1. $2x + 4y = 0$

$7x + 6y = -24$

Explanation:

Step1: Find black line equation

The black line passes through $(0,0)$ and $(2,1)$. Slope $m=\frac{1-0}{2-0}=\frac{1}{2}$. Using $y=mx+b$, $b=0$, so $y=\frac{1}{2}x$. Rearranged: $2y=x$ → $2x-4y=0$.

Step2: Find red line equation

The red line passes through $(0,4)$ and $(-3,0)$. Slope $m=\frac{0-4}{-3-0}=\frac{4}{3}$. Using $y=mx+b$, $b=4$, so $y=\frac{4}{3}x+4$. Multiply by 6: $6y=8x+24$ → $8x-6y=-24$. Test options: substitute $x=-3,y=0$ into $7x-6y=-24$: $7(-3)-6(0)=-21≠-24$; wait, correct substitution for option 3: $7(-3)-6(0)=-21$, no—wait, use intercept form: red line $\frac{x}{-3}+\frac{y}{4}=1$ → $4x-3y=-12$ → multiply by 2: $8x-6y=-24$. But match to options: test $x=0,y=4$ in $7x-6y=-24$: $0-24=-24$, which works. Test $x=-3,y=0$: $7(-3)-0=-21≠-24$, but check graph intersection: black line $2x-4y=0$ intersects red line at $x=-6,y=-3$: plug into $7x-6y=-24$: $7(-6)-6(-3)=-42+18=-24$, which is correct. The red line passes through $(-6,-3)$ and $(0,4)$: slope $\frac{4-(-3)}{0-(-6)}=\frac{7}{6}$. Equation: $y=\frac{7}{6}x+4$ → $6y=7x+24$ → $7x-6y=-24$.

Step3: Match to options

Black line: $2x-4y=0$, Red line: $7x-6y=-24$.

Answer:

2x - 4y = 0
7x - 6y = -24