Question

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the elimination method part 2: the equations strike back
solve each system using elimination. show all of your work. written checks are optional.
1)
4x - y = 6
3x + 2y = 21
2)
5x + 9y = 112
27x - 18y = 72

Explanation:

Problem 1:

Step 1: Label the equations

Let \( 4x - y = 6 \) be Equation (1) and \( 3x + 2y = 21 \) be Equation (2).

Step 2: Eliminate \( y \)

Multiply Equation (1) by 2: \( 2(4x - y) = 2\times6 \) gives \( 8x - 2y = 12 \) (Equation 3).

Step 3: Add Equation (2) and Equation (3)

\( (3x + 2y) + (8x - 2y) = 21 + 12 \)
Simplify: \( 3x + 8x + 2y - 2y = 33 \)
\( 11x = 33 \)

Step 4: Solve for \( x \)

Divide both sides by 11: \( x = \frac{33}{11} = 3 \)

Step 5: Substitute \( x = 3 \) into Equation (1)

\( 4(3) - y = 6 \)
\( 12 - y = 6 \)
Subtract 12 from both sides: \( -y = 6 - 12 = -6 \)
Multiply both sides by -1: \( y = 6 \)

Step 1: Label the equations

Let \( 5x + 9y = 112 \) be Equation (1) and \( 27x - 18y = 72 \) be Equation (2). Simplify Equation (2) by dividing by 9: \( 3x - 2y = 8 \) (Equation 3).

Step 2: Eliminate \( y \)

Multiply Equation (1) by 2: \( 2(5x + 9y) = 2\times112 \) gives \( 10x + 18y = 224 \) (Equation 4).
Multiply Equation (3) by 9: \( 9(3x - 2y) = 9\times8 \) gives \( 27x - 18y = 72 \) (Equation 5, same as original Equation 2, but we'll use Equation 4 and Equation 3 for elimination). Wait, better to use Equation (1) and Equation (3) to eliminate \( y \). Multiply Equation (3) by \( \frac{9}{2} \)? No, better: Multiply Equation (1) by 2: \( 10x + 18y = 224 \) (Equation 4), Multiply Equation (3) by 9: \( 27x - 18y = 72 \) (Equation 5). Wait, no, Equation (3) is \( 3x - 2y = 8 \), multiply by 9: \( 27x - 18y = 72 \) (same as Equation 2). Instead, multiply Equation (1) by 2: \( 10x + 18y = 224 \) (Equation 4), multiply Equation (3) by 9: \( 27x - 18y = 72 \) (Equation 5). Now add Equation (4) and Equation (5):

\( (10x + 18y) + (27x - 18y) = 224 + 72 \)
\( 37x = 296 \)

Step 3: Solve for \( x \)

\( x = \frac{296}{37} = 8 \)

Step 4: Substitute \( x = 8 \) into Equation (3)

\( 3(8) - 2y = 8 \)
\( 24 - 2y = 8 \)
Subtract 24: \( -2y = 8 - 24 = -16 \)
Divide by -2: \( y = \frac{-16}{-2} = 8 \)

Answer:

\( x = 3 \), \( y = 6 \)

Problem 2: