Question

solve the system,
\

$$\begin{cases} 3x + 2y + 4z = 11 \\ 2x - y + 3z = 4 \\ 5x - 3y + 5z = -1 \\end{cases}$$

enter your answer as an ordered triple (x, y, z).
(?, , )

Explanation:

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

First, take the second equation \( 2x - y + 3z = 4 \), multiply it by 2: \( 4x - 2y + 6z = 8 \).
Add it to the first equation \( 3x + 2y + 4z = 11 \):
\( (3x + 4x) + (2y - 2y) + (4z + 6z) = 11 + 8 \)
\( 7x + 10z = 19 \) (Equation 4)

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

Multiply the second equation \( 2x - y + 3z = 4 \) by 3: \( 6x - 3y + 9z = 12 \).
Subtract the third equation \( 5x - 3y + 5z = -1 \) from it:
\( (6x - 5x) + (-3y + 3y) + (9z - 5z) = 12 - (-1) \)
\( x + 4z = 13 \) (Equation 5)

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

From Equation 5: \( x = 13 - 4z \).
Substitute \( x = 13 - 4z \) into Equation 4:
\( 7(13 - 4z) + 10z = 19 \)
\( 91 - 28z + 10z = 19 \)
\( -18z = 19 - 91 \)
\( -18z = -72 \)
\( z = 4 \)

Step4: Find \( x \) using \( z = 4 \)

Substitute \( z = 4 \) into \( x = 13 - 4z \):
\( x = 13 - 4(4) = 13 - 16 = -3 \)

Step5: Find \( y \) using \( x = -3 \) and \( z = 4 \)

Substitute \( x = -3 \) and \( z = 4 \) into the second equation \( 2x - y + 3z = 4 \):
\( 2(-3) - y + 3(4) = 4 \)
\( -6 - y + 12 = 4 \)
\( 6 - y = 4 \)
\( -y = 4 - 6 = -2 \)
\( y = 2 \)

Answer:

\((-3, 2, 4)\)