Question

1. below is a system of equations.

$x + 4y = -4$
$2x + 8y = 16$
a. solve this system using elimination.
b. graph this system of equations.

Explanation:

Step1: Scale first equation by -2

Multiply $x + 4y = -4$ by $-2$:
$-2x - 8y = 8$

Step2: Add to second equation

Add to $2x + 8y = 16$:
$(-2x - 8y) + (2x + 8y) = 8 + 16$
$0 = 24$

Step3: Analyze the result

$0=24$ is a false statement, so no solution exists.

Step4: Rewrite equations for graphing

Rewrite $x + 4y = -4$ in slope-intercept form:
$4y = -x -4 \implies y = -\frac{1}{4}x -1$
Rewrite $2x + 8y = 16$ in slope-intercept form:
$8y = -2x +16 \implies y = -\frac{1}{4}x +2$

Step5: Identify key graph features

Both lines have slope $-\frac{1}{4}$ (parallel, no intersection).

• For $y = -\frac{1}{4}x -1$: y-intercept $(0, -1)$, x-intercept $(-4, 0)$
• For $y = -\frac{1}{4}x +2$: y-intercept $(0, 2)$, x-intercept $(8, 0)$

Answer:

A. No solution (the system is inconsistent, as elimination leads to a false statement $0=24$)
B.

1. Plot the line $y = -\frac{1}{4}x -1$ using points $(-4, 0)$ and $(0, -1)$, then draw a straight line through them.
2. Plot the line $y = -\frac{1}{4}x +2$ using points $(8, 0)$ and $(0, 2)$, then draw a straight line through them.

The two lines are parallel and never intersect, which matches the conclusion of no solution.