Question

s 1–2 solve each system of equations.

1. $-5x + y - 4z = 60$

$2x + 4y + 3z = -12$
$6x - 3y - 2z = -52$

Explanation:

Step1: Eliminate \( y \) from first two equations

First equation: \( -5x + y - 4z = 60 \) can be rewritten as \( y = 5x + 4z + 60 \).
Substitute \( y = 5x + 4z + 60 \) into the second equation \( 2x + 4y + 3z = -12 \):
\( 2x + 4(5x + 4z + 60) + 3z = -12 \)
\( 2x + 20x + 16z + 240 + 3z = -12 \)
\( 22x + 19z = -252 \) (Equation 4)

Step2: Eliminate \( y \) from first and third equations

Substitute \( y = 5x + 4z + 60 \) into the third equation \( 6x - 3y - 2z = -52 \):
\( 6x - 3(5x + 4z + 60) - 2z = -52 \)
\( 6x - 15x - 12z - 180 - 2z = -52 \)
\( -9x - 14z = 128 \) (Equation 5)

Step3: Solve Equations 4 and 5 for \( x \) and \( z \)

Multiply Equation 4 by 9: \( 198x + 171z = -2268 \) (Equation 6)
Multiply Equation 5 by 22: \( -198x - 308z = 2816 \) (Equation 7)
Add Equation 6 and Equation 7:
\( (198x - 198x) + (171z - 308z) = -2268 + 2816 \)
\( -137z = 548 \)
\( z = -4 \)

Substitute \( z = -4 \) into Equation 4: \( 22x + 19(-4) = -252 \)
\( 22x - 76 = -252 \)
\( 22x = -176 \)
\( x = -8 \)

Step4: Find \( y \) using \( y = 5x + 4z + 60 \)

Substitute \( x = -8 \), \( z = -4 \):
\( y = 5(-8) + 4(-4) + 60 \)
\( y = -40 - 16 + 60 \)
\( y = 4 \)

Answer:

\( x = -8 \), \( y = 4 \), \( z = -4 \) (or the solution set \( (-8, 4, -4) \))