Question

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solve each system of equations by elimination. clearly identify your solution.

1. $\begin{cases} y = -x + 1 \\\\ y = 4x - 14 end{cases}$
2. $\begin{cases} y = -3x + 5 \\\\ y = -8x + 25 end{cases}$
3. $\begin{cases} x - 2y = 10 \\\\ x + 3y = 5 end{cases}$
4. $\begin{cases} 2x - 3y = 9 \\\\ -5x - 3y = 30 end{cases}$
5. $\begin{cases} x + y = -4 \\\\ x - y = 2 end{cases}$
6. $\begin{cases} 2x - 3y = 14 \\\\ x + 3y = -11 end{cases}$
7. $\begin{cases} -3x - 4y = -1 \\\\ 3x - y = -4 end{cases}$
8. $\begin{cases} 6x + 5y = 4 \\\\ 6x - 7y = -20 end{cases}$

© gina wilson (all things algebra®, llc), 2012-2016

Explanation:

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Problem 1

Step1: Set equations equal

$-x + 1 = 4x - 14$

Step2: Isolate $x$ terms

$1 + 14 = 4x + x$
$15 = 5x$

Step3: Solve for $x$

$x = \frac{15}{5} = 3$

Step4: Substitute $x$ to find $y$

$y = -(3) + 1 = -2$

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Problem 2

Step1: Set equations equal

$-3x + 5 = -8x + 25$

Step2: Isolate $x$ terms

$-3x + 8x = 25 - 5$
$5x = 20$

Step3: Solve for $x$

$x = \frac{20}{5} = 4$

Step4: Substitute $x$ to find $y$

$y = -3(4) + 5 = -7$

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Problem 3

Step1: Subtract equations to eliminate $x$

$(x - 2y) - (x + 3y) = 10 - 5$
$-5y = 5$

Step2: Solve for $y$

$y = \frac{5}{-5} = -1$

Step3: Substitute $y$ to find $x$

$x - 2(-1) = 10$
$x + 2 = 10$
$x = 8$

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Problem 4

Step1: Subtract equations to eliminate $y$

$(2x - 3y) - (-5x - 3y) = 9 - 30$
$7x = -21$

Step2: Solve for $x$

$x = \frac{-21}{7} = -3$

Step3: Substitute $x$ to find $y$

$2(-3) - 3y = 9$
$-6 - 3y = 9$
$-3y = 15$
$y = -5$

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Problem 5

Step1: Add equations to eliminate $y$

$(x + y) + (x - y) = -4 + 2$
$2x = -2$

Step2: Solve for $x$

$x = \frac{-2}{2} = -1$

Step3: Substitute $x$ to find $y$

$-1 + y = -4$
$y = -3$

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Problem 6

Step1: Add equations to eliminate $y$

$(2x - 3y) + (x + 3y) = 14 + (-11)$
$3x = 3$

Step2: Solve for $x$

$x = \frac{3}{3} = 1$

Step3: Substitute $x$ to find $y$

$1 + 3y = -11$
$3y = -12$
$y = -4$

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Problem 7

Step1: Add equations to eliminate $x$

$(-3x - 4y) + (3x - y) = -1 + (-4)$
$-5y = -5$

Step2: Solve for $y$

$y = \frac{-5}{-5} = 1$

Step3: Substitute $y$ to find $x$

$3x - 1 = -4$
$3x = -3$
$x = -1$

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Problem 8

Step1: Subtract equations to eliminate $x$

$(6x + 5y) - (6x - 7y) = 4 - (-20)$
$12y = 24$

Step2: Solve for $y$

$y = \frac{24}{12} = 2$

Step3: Substitute $y$ to find $x$

$6x + 5(2) = 4$
$6x + 10 = 4$
$6x = -6$
$x = -1$

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Answer:

1. $(3, -2)$
2. $(4, -7)$
3. $(8, -1)$
4. $(-3, -5)$
5. $(-1, -3)$
6. $(1, -4)$
7. $(-1, 1)$
8. $(-1, 2)$