To solve the system of linear equations
[tex]\[ 2x + 3y = 12 \][/tex]
[tex]\[ x - y = 2 \][/tex]
we can use the method of substitution or elimination. Here, we'll use the substitution method first:
1. Solve the second equation for [tex]\(x\)[/tex]:
[tex]\[ x = y + 2 \][/tex]
2. Substitute this expression for [tex]\(x\)[/tex] into the first equation:
[tex]\[ 2(y + 2) + 3y = 12 \][/tex]
3. Distribute and simplify:
[tex]\[ 2y + 4 + 3y = 12 \][/tex]
[tex]\[ 5y + 4 = 12 \][/tex]
4. Subtract 4 from both sides:
[tex]\[ 5y = 8 \][/tex]
5. Divide by 5:
[tex]\[ y = \frac{8}{5} \][/tex]
Now that we have the value of [tex]\(y\)[/tex], we can find [tex]\(x\)[/tex] by substituting [tex]\(y = \frac{8}{5}\)[/tex] back into the expression [tex]\(x = y + 2\)[/tex]:
[tex]\[ x = \frac{8}{5} + 2 \][/tex]
6. Convert 2 to a fraction with a common denominator of 5:
[tex]\[ 2 = \frac{10}{5} \][/tex]
7. Add the fractions:
[tex]\[ x = \frac{8}{5} + \frac{10}{5} \][/tex]
[tex]\[ x = \frac{18}{5} \][/tex]
So, the solution to the system of equations is:
[tex]\[ x = \frac{18}{5} \][/tex]
[tex]\[ y = \frac{8}{5} \][/tex]
