Answer :
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]
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]
