Question

1. \

$$\begin{cases} x + 3y + 10z - 13 = 0, \\\\ 2x - 2y + 3z - 16.5 = 0, \\\\ 3x - y + 4z - 20 = 0 \\end{cases}$$

Explanation:

Step1: Label the equations

Let's label the equations for clarity:

• Equation (1): \( x + 3y + 10z - 13 = 0 \) (rewritten to standard form)
• Equation (2): \( 2x - 2y + 3z - 16.5 = 0 \)
• Equation (3): \( 3x - y + 4z - 20 = 0 \)

Step2: Eliminate \( x \) from Equation (2) and Equation (1)

Multiply Equation (1) by 2: \( 2x + 6y + 20z - 26 = 0 \) (let's call this Equation (4))
Subtract Equation (2) from Equation (4):
\( (2x + 6y + 20z - 26) - (2x - 2y + 3z - 16.5) = 0 - 0 \)
Simplify: \( 8y + 17z - 9.5 = 0 \) (Equation (5))

Step3: Eliminate \( x \) from Equation (3) and Equation (1)

Multiply Equation (1) by 3: \( 3x + 9y + 30z - 39 = 0 \) (Equation (6))
Subtract Equation (3) from Equation (6):
\( (3x + 9y + 30z - 39) - (3x - y + 4z - 20) = 0 - 0 \)
Simplify: \( 10y + 26z - 19 = 0 \) (Equation (7))

Step4: Solve Equations (5) and (7) for \( y \) and \( z \)

Multiply Equation (5) by 5: \( 40y + 85z - 47.5 = 0 \) (Equation (8))
Multiply Equation (7) by 4: \( 40y + 104z - 76 = 0 \) (Equation (9))
Subtract Equation (8) from Equation (9):
\( (40y + 104z - 76) - (40y + 85z - 47.5) = 0 - 0 \)
Simplify: \( 19z - 28.5 = 0 \)
Solve for \( z \): \( 19z = 28.5 \) → \( z = \frac{28.5}{19} = 1.5 \)

Step5: Substitute \( z = 1.5 \) into Equation (5) to find \( y \)

\( 8y + 17(1.5) - 9.5 = 0 \)
\( 8y + 25.5 - 9.5 = 0 \)
\( 8y + 16 = 0 \) → \( 8y = -16 \) → \( y = -2 \)

Step6: Substitute \( y = -2 \) and \( z = 1.5 \) into Equation (1) to find \( x \)

\( x + 3(-2) + 10(1.5) - 13 = 0 \)
\( x - 6 + 15 - 13 = 0 \)
\( x - 4 = 0 \) → \( x = 4 \)

Answer:

\( x = 4 \), \( y = -2 \), \( z = 1.5 \) (or \( z = \frac{3}{2} \))