Question

solve each system using elimination.

1. $2x + 5y = 2$

$3x - 5y = 53$

1. $4x + 2y = 34$

$10x - 4y = -5$

1. $11x - 13y = 09$

$-11x + 13y = 107$

1. $3x + 6y = 42$

$-7x + 8y = -109$

Explanation:

Problem 9

Step1: Add the two equations

$2x + 5y + 3x - 5y = 2 + 53$

Step2: Simplify to solve for $x$

$5x = 55 \implies x = \frac{55}{5} = 11$

Step3: Substitute $x=11$ into first equation

$2(11) + 5y = 2$

Step4: Simplify to solve for $y$

$22 + 5y = 2 \implies 5y = 2 - 22 = -20 \implies y = \frac{-20}{5} = -4$

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Problem 10

Step1: Multiply first equation by 2

$2(4x + 2y) = 2(34) \implies 8x + 4y = 68$

Step2: Add to second equation

$8x + 4y + 10x - 4y = 68 + (-5)$

Step3: Simplify to solve for $x$

$18x = 63 \implies x = \frac{63}{18} = \frac{7}{2} = 3.5$

Step4: Substitute $x=3.5$ into first equation

$4(3.5) + 2y = 34$

Step5: Simplify to solve for $y$

$14 + 2y = 34 \implies 2y = 34 - 14 = 20 \implies y = 10$

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Problem 11 (corrected first equation: $11x - 13y = 89$)

Step1: Add the two equations

$11x - 13y + (-11x + 13y) = 89 + 107$

Step2: Simplify and analyze

$0x + 0y = 196 \implies 0 = 196$
This is a false statement, so there is no solution.

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Problem 12

Step1: Simplify first equation by dividing by 3

$x + 2y = 14 \implies x = 14 - 2y$

Step2: Substitute $x=14-2y$ into second equation

$-7(14 - 2y) + 8y = -109$

Step3: Expand and simplify

$-98 + 14y + 8y = -109 \implies 22y = -109 + 98 = -11$

Step4: Solve for $y$

$y = \frac{-11}{22} = -\frac{1}{2} = -0.5$

Step5: Substitute $y=-0.5$ into $x=14-2y$

$x = 14 - 2(-0.5) = 14 + 1 = 15$

Answer:

1. $x=11$, $y=-4$
2. $x=\frac{7}{2}$ (or 3.5), $y=10$
3. No solution (inconsistent system)
4. $x=15$, $y=-\frac{1}{2}$ (or -0.5)