Answer :

GinnyAnswer
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]