To solve this system of linear equations step-by-step, we aim to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations:
[tex]\[
\left\{\begin{array}{l}
5x - 6y + z = 49 \quad \text{(1)} \\
-x + 5y + 3z = -14 \quad \text{(2)} \\
2x - 2y + 2z = 28 \quad \text{(3)}
\end{array}\right.
\][/tex]
Step 1: Simplify Equation (3)
Divide all terms in equation (3) by 2:
[tex]\[
x - y + z = 14 \quad \text{(4)}
\][/tex]
Now our system looks like:
[tex]\[
\left\{\begin{array}{l}
5x - 6y + z = 49 \quad \text{(1)} \\
-x + 5y + 3z = -14 \quad \text{(2)} \\
x - y + z = 14 \quad \text{(4)}
\end{array}\right.
\][/tex]
Step 2: Eliminate one variable
First, let's eliminate [tex]\( z \)[/tex] between equations (1) and (4) by subtracting (4) from (1).
[tex]\[
(5x - 6y + z) - (x - y + z) = 49 - 14
\][/tex]
Simplify the left side:
[tex]\[
4x - 5y = 35 \quad \text{(5)}
\][/tex]
Next, eliminate [tex]\( z \)[/tex] between equations (2) and (4). Multiply equation (4) by 3 to line up like terms with equation (2):
[tex]\[
3(x - y + z) = 3(14) \implies 3x - 3y + 3z = 42
\][/tex]
Subtract this equation from equation (2):
[tex]\[
(-x + 5y + 3z) - (3x - 3y + 3z) = -14 - 42
\][/tex]
Simplify the left side:
[tex]\[
-4x + 8y = -56 \quad \text{(6)}
\][/tex]
Step 3: Simplify and Solve the New System
We now have a simpler system with two variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
\left\{\begin{array}{l}
4x - 5y = 35 \quad \text{(5)} \\
-4x + 8y = -56 \quad \text{(6)}
\end{array}\right.
\][/tex]
Add equations (5) and (6) to eliminate [tex]\( x \)[/tex]:
[tex]\[
(4x - 5y) + (-4x + 8y) = 35 - 56
\][/tex]
This simplifies to:
[tex]\[
3y = -21
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = -7
\][/tex]
Step 4: Substitute [tex]\( y \)[/tex] back into one of the simplified equations
Using equation (5):
[tex]\[
4x - 5(-7) = 35 \implies 4x + 35 = 35
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
4x = 0 \implies x = 0
\][/tex]
Step 5: Substitute [tex]\( x \)[/tex] and [tex]\( y \)[/tex] back into one of the original equations
Use equation (4) to solve for [tex]\( z \)[/tex]:
[tex]\[
0 - (-7) + z = 14 \implies 7 + z = 14
\][/tex]
Solving for [tex]\( z \)[/tex]:
[tex]\[
z = 7
\][/tex]
Final Answer:
The solution to the system of equations is:
[tex]\[
x = 0, \quad y = -7, \quad z = 7
\][/tex]
Thus, the values that satisfy the original system of equations are [tex]\( x = 0 \)[/tex], [tex]\( y = -7 \)[/tex], and [tex]\( z = 7 \)[/tex].
