Question

$2x + 9y = 27$
$x - 3y = -24$
$x - 2y = 8$
$8x + 6y = 42$
$2x - 2y = 14$
$x - y = 2$
$x = 4y$
$x + 2y = 12$
© gina wilson (all things algebra®, llc), 2012-2016

Explanation:

System 1:

$$\begin{cases}2x + 9y = 27 \\ x - 3y = -24\end{cases}$$

Step1: Eliminate $x$, multiply eq2 by 2

$2(x - 3y) = 2(-24) \implies 2x - 6y = -48$

Step2: Subtract new eq2 from eq1

$(2x + 9y) - (2x - 6y) = 27 - (-48)$
$15y = 75$

Step3: Solve for $y$

$y = \frac{75}{15} = 5$

Step4: Substitute $y=5$ into eq2

$x - 3(5) = -24 \implies x - 15 = -24 \implies x = -9$

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System 2:

$$\begin{cases}x - 2y = 8 \\ 8x + 6y = 42\end{cases}$$

Step1: Eliminate $x$, multiply eq1 by 8

$8(x - 2y) = 8(8) \implies 8x - 16y = 64$

Step2: Subtract new eq1 from eq2

$(8x + 6y) - (8x - 16y) = 42 - 64$
$22y = -22$

Step3: Solve for $y$

$y = \frac{-22}{22} = -1$

Step4: Substitute $y=-1$ into eq1

$x - 2(-1) = 8 \implies x + 2 = 8 \implies x = 6$

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System 3:

$$\begin{cases}2x - 2y = 14 \\ x - y = 2\end{cases}$$

Step1: Simplify first equation

Divide eq1 by 2: $x - y = 7$

Step2: Compare with second equation

We have $x - y = 7$ and $x - y = 2$, which are parallel, inconsistent lines.

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System 4:

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

Step1: Substitute $x=4y$ into eq2

$4y + 2y = 12$

Step2: Solve for $y$

$6y = 12 \implies y = \frac{12}{6} = 2$

Step3: Substitute $y=2$ into $x=4y$

$x = 4(2) = 8$

Answer:

1. $x=-9, y=5$
2. $x=6, y=-1$
3. No solution (parallel lines)
4. $x=8, y=2$