Question

solve the following system of linear equations to find the value
$-x - y + 5z = 10$
$x + 3y + 3z = 2$
$-2x - 3y - z = 6$
options:
a. $x = 4$
b. $y = -4$
c. $z = 3$
d. $x = -3$
e. $z = 2$
f. none of the above

Explanation:

Step1: Identify the system of equations

We have the system:

$$
LATEXBLOCK0
$$

Step2: Add the first and second equations

Adding the first equation \(-x - y + 5z = 10\) and the second equation \(x + 3y + 3z = 2\) to eliminate \(x\):

$$
LATEXBLOCK1
$$

Step3: Multiply the second equation by 2 and add to the third equation

Multiply the second equation \(x + 3y + 3z = 2\) by 2: \(2x+6y + 6z=4\)
Add it to the third equation \(-2x - 3y - z = 6\):

$$
LATEXBLOCK2
$$

Step4: Solve Equation 4 for y and substitute into Equation 5

From Equation 4: \(y=6 - 4z\)
Substitute into Equation 5:

$$
LATEXBLOCK3
$$

Step5: Substitute z into Equation 4 to find y

Substitute \(z = \frac{8}{7}\) into \(y=6 - 4z\):

$$
LATEXBLOCK4
$$

Step6: Substitute y and z into the second equation to find x

Substitute \(y=\frac{10}{7}\) and \(z=\frac{8}{7}\) into \(x + 3y + 3z = 2\):

$$
LATEXBLOCK5
$$

Answer:

\(x = -\frac{40}{7}\), \(y=\frac{10}{7}\), \(z=\frac{8}{7}\)