Question

date: __________ bell: ____ homework 1: solving systems by graphing
this is a 2-page document
solve each system of equations by graphing. clearly identify your solution.

1. $\

$$\begin{cases} y=\\frac{2}{3}x-1 \\\\ y=-x+4 \\end{cases}$$

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1. $\

$$\begin{cases} x-y=7 \\\\ x-y=-4 \\end{cases}$$

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1. $\

$$\begin{cases} y=x-1 \\\\ x+4y=16 \\end{cases}$$

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1. $\

$$\begin{cases} 5x+2y=4 \\\\ 9x+2y=12 \\end{cases}$$

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1. $\

$$\begin{cases} 2x-6y=30 \\\\ y=\\frac{1}{3}x+1 \\end{cases}$$

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1. $\

$$\begin{cases} 5x+4y=-12 \\\\ 3x-4y=-20 \\end{cases}$$

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Explanation:

Step1: Solve system 1 algebraically

Set equations equal:
$\frac{2}{3}x - 1 = -x + 4$
Multiply by 3: $2x - 3 = -3x + 12$
$5x = 15 \implies x=3$
Substitute $x=3$: $y = -3 + 4 = 1$

Step2: Analyze system 2

Rewrite equations: $y = x - 7$, $y = x + 4$
Parallel lines (same slope, different intercepts)

Step3: Solve system 3 algebraically

Substitute $y=x-1$ into $x+4y=16$:
$x + 4(x-1) = 16$
$x + 4x - 4 = 16$
$5x = 20 \implies x=4$
Substitute $x=4$: $y=4-1=3$

Step4: Solve system 4 algebraically

Subtract first equation from second:
$(9x+2y)-(5x+2y)=12-4$
$4x=8 \implies x=2$
Substitute $x=2$: $5(2)+2y=4 \implies 10+2y=4 \implies 2y=-6 \implies y=-3$

Step5: Solve system 5 algebraically

Rewrite first equation: $2x-6y=30 \implies y=\frac{1}{3}x - 5$
Compare to $y=\frac{1}{3}x +1$: parallel lines

Step6: Solve system 6 algebraically

Add equations:
$(5x+4y)+(3x-4y)=-12+(-20)$
$8x=-32 \implies x=-4$
Substitute $x=-4$: $5(-4)+4y=-12 \implies -20+4y=-12 \implies 4y=8 \implies y=2$

Answer:

1. Solution: $(3, 1)$
2. No solution (parallel lines)
3. Solution: $(4, 3)$
4. Solution: $(2, -3)$
5. No solution (parallel lines)
6. Solution: $(-4, 2)$