Answer :
To solve the system of equations using the addition method, we need to eliminate one of the variables by adding the equations together. Here are the given equations:
[tex]\[ \begin{array}{l} 4x - 5y = -27 \\ 3x + 5y = 6 \end{array} \][/tex]
### Step-by-Step Solution:
1. Write down the system of equations:
[tex]\[ 4x - 5y = -27 \quad \text{(Equation 1)} \][/tex]
[tex]\[ 3x + 5y = 6 \quad \text{(Equation 2)} \][/tex]
2. Add the two equations to eliminate [tex]\( y \)[/tex]:
[tex]\[ (4x - 5y) + (3x + 5y) = -27 + 6 \][/tex]
3. Combine like terms:
[tex]\[ 4x + 3x - 5y + 5y = -27 + 6 \][/tex]
[tex]\[ 7x = -21 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-21}{7} = -3 \][/tex]
5. Substitute [tex]\( x = -3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
We use Equation 2:
[tex]\[ 3x + 5y = 6 \][/tex]
Substituting [tex]\( x = -3 \)[/tex]:
[tex]\[ 3(-3) + 5y = 6 \][/tex]
[tex]\[ -9 + 5y = 6 \][/tex]
6. Isolate [tex]\( y \)[/tex]:
[tex]\[ 5y = 6 + 9 \][/tex]
[tex]\[ 5y = 15 \][/tex]
[tex]\[ y = \frac{15}{5} = 3 \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ (x, y) = (-3, 3) \][/tex]
The solution set is [tex]\((-3, 3)\)[/tex].
[tex]\[ \begin{array}{l} 4x - 5y = -27 \\ 3x + 5y = 6 \end{array} \][/tex]
### Step-by-Step Solution:
1. Write down the system of equations:
[tex]\[ 4x - 5y = -27 \quad \text{(Equation 1)} \][/tex]
[tex]\[ 3x + 5y = 6 \quad \text{(Equation 2)} \][/tex]
2. Add the two equations to eliminate [tex]\( y \)[/tex]:
[tex]\[ (4x - 5y) + (3x + 5y) = -27 + 6 \][/tex]
3. Combine like terms:
[tex]\[ 4x + 3x - 5y + 5y = -27 + 6 \][/tex]
[tex]\[ 7x = -21 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-21}{7} = -3 \][/tex]
5. Substitute [tex]\( x = -3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
We use Equation 2:
[tex]\[ 3x + 5y = 6 \][/tex]
Substituting [tex]\( x = -3 \)[/tex]:
[tex]\[ 3(-3) + 5y = 6 \][/tex]
[tex]\[ -9 + 5y = 6 \][/tex]
6. Isolate [tex]\( y \)[/tex]:
[tex]\[ 5y = 6 + 9 \][/tex]
[tex]\[ 5y = 15 \][/tex]
[tex]\[ y = \frac{15}{5} = 3 \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ (x, y) = (-3, 3) \][/tex]
The solution set is [tex]\((-3, 3)\)[/tex].