Sure, let's solve the system of linear equations step-by-step:
[tex]\[
\begin{cases}
-x + y = 2 \\
3x + y = 10
\end{cases}
\][/tex]
Step 1: Express [tex]\( y \)[/tex] from the first equation.
From the first equation:
[tex]\[ -x + y = 2 \][/tex]
we can isolate [tex]\( y \)[/tex] by adding [tex]\( x \)[/tex] to both sides:
[tex]\[ y = x + 2 \][/tex]
Step 2: Substitute the expression for [tex]\( y \)[/tex] in the second equation.
Next, we substitute [tex]\( y = x + 2 \)[/tex] into the second equation:
[tex]\[ 3x + y = 10 \][/tex]
Replacing [tex]\( y \)[/tex] with [tex]\( x + 2 \)[/tex]:
[tex]\[ 3x + (x + 2) = 10 \][/tex]
Step 3: Solve for [tex]\( x \)[/tex].
Combine like terms:
[tex]\[ 3x + x + 2 = 10 \][/tex]
[tex]\[ 4x + 2 = 10 \][/tex]
Subtract 2 from both sides:
[tex]\[ 4x = 8 \][/tex]
Divide both sides by 4:
[tex]\[ x = 2 \][/tex]
Step 4: Find [tex]\( y \)[/tex] using the value of [tex]\( x \)[/tex].
Using the expression [tex]\( y = x + 2 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 2 + 2 \][/tex]
[tex]\[ y = 4 \][/tex]
Step 5: Verify the solution.
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 4 \)[/tex] back into the original equations to ensure the solution is correct.
First equation:
[tex]\[ -x + y = 2 \][/tex]
[tex]\[ -2 + 4 = 2 \][/tex]
[tex]\[ 2 = 2 \][/tex] (True)
Second equation:
[tex]\[ 3x + y = 10 \][/tex]
[tex]\[ 3(2) + 4 = 10 \][/tex]
[tex]\[ 6 + 4 = 10 \][/tex]
[tex]\[ 10 = 10 \][/tex] (True)
Both equations are satisfied, so the solution is correct.
Thus, the solution to the system of equations is:
[tex]\[ x = 2 \][/tex]
[tex]\[ y = 4 \][/tex]
Summarized solution: [tex]\( \boxed{(2, 4)} \)[/tex]
