Certainly! Let's solve the following system of linear equations step-by-step:
[tex]\[
\begin{cases}
2x + 3y = 10 \\
x + y = 12
\end{cases}
\][/tex]
1. Express one variable in terms of the other using the simpler equation:
We can start with the second equation:
[tex]\[
x + y = 12
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y = 12 - x
\][/tex]
2. Substitute this expression into the first equation:
Now, substitute [tex]\(y = 12 - x\)[/tex] into the first equation:
[tex]\[
2x + 3(12 - x) = 10
\][/tex]
3. Simplify and solve for [tex]\(x\)[/tex]:
Distribute the 3:
[tex]\[
2x + 36 - 3x = 10
\][/tex]
Combine like terms:
[tex]\[
2x - 3x + 36 = 10 \implies -x + 36 = 10
\][/tex]
Isolate [tex]\(x\)[/tex] by subtracting 36 from both sides:
[tex]\[
-x = 10 - 36 \implies -x = -26
\][/tex]
Multiply both sides by -1:
[tex]\[
x = 26
\][/tex]
4. Substitute the value of [tex]\(x\)[/tex] back into the expression for [tex]\(y\)[/tex]:
Recall from step 1 that [tex]\(y = 12 - x\)[/tex]:
[tex]\[
y = 12 - 26
\][/tex]
Simplify:
[tex]\[
y = -14
\][/tex]
5. Verify the solution:
Substitute [tex]\(x = 26\)[/tex] and [tex]\(y = -14\)[/tex] into both original equations to ensure that they are satisfied:
[tex]\[
2x + 3y = 2(26) + 3(-14) = 52 - 42 = 10 \quad \text{(True)}
\][/tex]
[tex]\[
x + y = 26 + (-14) = 26 - 14 = 12 \quad \text{(True)}
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
\boxed{x = 26, y = -14}
\][/tex]
