Question

use a graphing calculator to solve the system.
$5x + y - 3z = 11$
$2x + 2y + z = 6$
$-2x - 4y + 4z = -8$
the solution for the system is $x = \square$, $y = \square$, and $z = \square$.

Explanation:

Step1: Write the system of equations

We have the system:

$$
LATEXBLOCK0
$$

Step2: Simplify the third equation

Divide the third equation by -2: \(x + 2y - 2z = 4\)
Now we have:

$$
LATEXBLOCK1
$$

Step3: Eliminate \(y\) from the second and third equations

Subtract the third equation from the second equation:
\((2x + 2y + z) - (x + 2y - 2z) = 6 - 4\)
\(2x + 2y + z - x - 2y + 2z = 2\)
\(x + 3z = 2\) --- Equation (4)

Step4: Eliminate \(y\) from the first and third equations

Multiply the third equation by \(\frac{1}{2}\): \(\frac{1}{2}x + y - z = 2\)
Subtract this from the first equation:
\((5x + y - 3z) - (\frac{1}{2}x + y - z) = 11 - 2\)
\(5x + y - 3z - \frac{1}{2}x - y + z = 9\)
\(\frac{9}{2}x - 2z = 9\)
Multiply by 2: \(9x - 4z = 18\) --- Equation (5)

Step5: Solve Equation (4) and Equation (5)

From Equation (4): \(x = 2 - 3z\)
Substitute \(x = 2 - 3z\) into Equation (5):
\(9(2 - 3z) - 4z = 18\)
\(18 - 27z - 4z = 18\)
\(18 - 31z = 18\)
\(-31z = 0\)
\(z = 0\)

Step6: Find \(x\)

Substitute \(z = 0\) into Equation (4): \(x + 3(0) = 2\) so \(x = 2\)

Step7: Find \(y\)

Substitute \(x = 2\) and \(z = 0\) into the second original equation: \(2(2) + 2y + 0 = 6\)
\(4 + 2y = 6\)
\(2y = 2\)
\(y = 1\)

Answer:

\(x = 2\), \(y = 1\), \(z = 0\)