Question

solve each system using elimination.

1. $2x + 5y = 2$

$3x - 5y = 53$

1. $4x + 2y = 34$

$10x - 4y = -5$

1. $11x - 13y = 09$

$-11x + 13y = 107$

Explanation:

Step1: Add equations to eliminate $y$

For problem 9:
$2x + 5y + 3x - 5y = 2 + 53$
$5x = 55$

Step2: Solve for $x$

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

Step3: Substitute $x=11$ to find $y$

$2(11) + 5y = 2$
$22 + 5y = 2$
$5y = 2 - 22 = -20$
$y = \frac{-20}{5} = -4$

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Step4: Scale first equation for $y$ elimination

For problem 10:
Multiply $4x + 2y = 34$ by 2: $8x + 4y = 68$

Step5: Add equations to eliminate $y$

$8x + 4y + 10x - 4y = 68 + (-5)$
$18x = 63$

Step6: Solve for $x$

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

Step7: Substitute $x=3.5$ to find $y$

$4(3.5) + 2y = 34$
$14 + 2y = 34$
$2y = 34 - 14 = 20$
$y = \frac{20}{2} = 10$

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Step8: Add equations to check consistency

For problem 11 (correcting first equation to $11x - 13y = 9$):
$11x - 13y + (-11x + 13y) = 9 + 107$
$0 = 116$

Answer:

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