To solve the system of equations using the substitution method, we start by expressing one variable in terms of the other using one of the given equations. In this case, equation 2 provides an expression for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
1. Write equation 2:
[tex]\[
y = x - 1
\][/tex]
2. Substitute this expression for [tex]\( y \)[/tex] into equation 1, which is:
[tex]\[
2x + 3y = 12
\][/tex]
Substituting [tex]\( y = x - 1 \)[/tex] into equation 1:
[tex]\[
2x + 3(x - 1) = 12
\][/tex]
3. Distribute [tex]\( 3 \)[/tex] inside the parentheses:
[tex]\[
2x + 3x - 3 = 12
\][/tex]
4. Combine like terms:
[tex]\[
5x - 3 = 12
\][/tex]
5. Solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex]:
[tex]\[
5x - 3 + 3 = 12 + 3
\][/tex]
[tex]\[
5x = 15
\][/tex]
[tex]\[
x = \frac{15}{5}
\][/tex]
[tex]\[
x = 3
\][/tex]
6. Now that we have [tex]\( x \)[/tex], use equation 2 to find [tex]\( y \)[/tex]:
[tex]\[
y = x - 1
\][/tex]
Substitute [tex]\( x = 3 \)[/tex]:
[tex]\[
y = 3 - 1
\][/tex]
[tex]\[
y = 2
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
(x, y) = (3, 2)
\][/tex]
