Question

math 8
u7 l3 solving systems of equations graphically
workspace:
use the space below to solve any equation for y.
a. $\begin{cases} -8x + 2y = 8 \\\\ y = 4x + 4 end{cases}$
b. $\begin{cases} y = -2x + 2 \\\\ 2x + y = -2 end{cases}$
c. $\begin{cases} y = \frac{1}{3}x - 3 \\\\ y = -x + 1 end{cases}$
d. $\begin{cases} x + 2y = 4 \\\\ y = -\frac{1}{2}x + 2 end{cases}$
e. $\begin{cases} y = -\frac{2}{3}x - 3 \\\\ y = -\frac{2}{3}x + 5 end{cases}$
f. $\begin{cases} y = \frac{3}{2}x - 2 \\\\ y = x - 2 end{cases}$
g. $\begin{cases} x + 3y = 9 \\\\ x + 3y = -3 end{cases}$
h. $\begin{cases} x + y = 2 \\\\ 2x + 2y = 4 end{cases}$
i. $\begin{cases} y = \frac{2}{3}x - 2 \\\\ y = x - 1 end{cases}$

Explanation:

Step1: Rearrange A first equation for $y$

$$\begin{align*} -8x + 2y &= 8 \\ 2y &= 8x + 8 \\ y &= 4x + 4 \end{align*}$$

Step2: Solve system A (substitute)

Second equation is $y=4x+4$, so infinitely many solutions: $y=4x+4$

Step3: Rearrange B second equation for $y$

$$\begin{align*} 2x + y &= -2 \\ y &= -2x -2 \end{align*}$$

Step4: Solve system B (set equal)

$$\begin{align*} -2x + 2 &= -2x -2 \\ 2 &= -2 \end{align*}$$

No solution.

Step5: Solve system C (set equal)

$$\begin{align*} \frac{1}{3}x - 3 &= -x + 1 \\ \frac{1}{3}x + x &= 1 + 3 \\ \frac{4}{3}x &= 4 \\ x &= 3 \end{align*}$$

Substitute $x=3$: $y = -3 + 1 = -2$

Step6: Rearrange D first equation for $y$

$$\begin{align*} x + 2y &= 4 \\ 2y &= -x + 4 \\ y &= -\frac{1}{2}x + 2 \end{align*}$$

Step7: Solve system D (substitute)

Second equation is $y=-\frac{1}{2}x+2$, so infinitely many solutions: $y=-\frac{1}{2}x+2$

Step8: Solve system E (set equal)

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

No solution.

Step9: Solve system F (set equal)

$$\begin{align*} \frac{3}{2}x - 2 &= x - 2 \\ \frac{3}{2}x - x &= 0 \\ \frac{1}{2}x &= 0 \\ x &= 0 \end{align*}$$

Substitute $x=0$: $y = 0 - 2 = -2$

Step10: Rearrange G equations for $y$

$$\begin{align*} x + 3y &= 9 \implies y = -\frac{1}{3}x + 3 \\ x + 3y &= -3 \implies y = -\frac{1}{3}x - 1 \end{align*}$$

Step11: Solve system G (compare)

Same slope, different intercepts: No solution.

Step12: Rearrange H second equation for $y$

$$\begin{align*} 2x + 2y &= 4 \\ y &= -x + 2 \end{align*}$$

Step13: Solve system H (set equal)

$$\begin{align*} x + y &= 2 \implies y = -x + 2 \end{align*}$$

Infinitely many solutions: $y=-x+2$

Step14: Solve system I (set equal)

$$\begin{align*} \frac{2}{3}x - 2 &= x - 1 \\ \frac{2}{3}x - x &= 2 - 1 \\ -\frac{1}{3}x &= 1 \\ x &= -3 \end{align*}$$

Substitute $x=-3$: $y = -3 - 1 = -4$

Answer:

A. Infinitely many solutions: $y = 4x + 4$
B. No solution
C. $(3, -2)$
D. Infinitely many solutions: $y = -\frac{1}{2}x + 2$
E. No solution
F. $(0, -2)$
G. No solution
H. Infinitely many solutions: $y = -x + 2$
I. $(-3, -4)$