Question

9
4a + 5b - 6c = 2
-3a - 2b + 7c = -15
-a + 4b + 2c = -13

1. -2x + 15y + z = 44

Explanation:

Step1: Label the equations

Let the equations be:
\(

$$\begin{cases} 4a + 5b - 6c = 2 & (1)\\ -3a - 2b + 7c = -15 & (2)\\ -a + 4b + 2c = -13 & (3) \end{cases}$$

\)

Step2: Eliminate \(a\) from (1) and (2), (1) and (3)

• Multiply (3) by 4: \( -4a + 16b + 8c = -52 \) (4)
• Add (1) and (4): \( (4a + 5b - 6c) + (-4a + 16b + 8c) = 2 + (-52) \)

\( 21b + 2c = -50 \) (5)

• Multiply (3) by 3: \( -3a + 12b + 6c = -39 \) (6)
• Subtract (2) from (6): \( (-3a + 12b + 6c) - (-3a - 2b + 7c) = -39 - (-15) \)

\( 14b - c = -24 \) (7)

Step3: Solve for \(b\) and \(c\) from (5) and (7)

• Multiply (7) by 2: \( 28b - 2c = -48 \) (8)
• Add (5) and (8): \( (21b + 2c) + (28b - 2c) = -50 + (-48) \)

\( 49b = -98 \) \( \Rightarrow b = -2 \)

• Substitute \(b = -2\) into (7): \( 14(-2) - c = -24 \)

\( -28 - c = -24 \) \( \Rightarrow c = -4 \)

Step4: Solve for \(a\) using (3)

Substitute \(b = -2\), \(c = -4\) into (3): \( -a + 4(-2) + 2(-4) = -13 \)
\( -a - 8 - 8 = -13 \) \( -a - 16 = -13 \) \( -a = 3 \) \( \Rightarrow a = -3 \)

Answer:

\( a = -3 \), \( b = -2 \), \( c = -4 \)