Answer :
To solve the given system of equations:
[tex]\[ \begin{cases} -x + 3y = 32 \\ 7x + 3y = -8 \end{cases} \][/tex]
we can use the method of elimination to find [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
1. Eliminate variable [tex]\( y \)[/tex]:
To eliminate [tex]\( y \)[/tex], we subtract the first equation from the second equation. This gives us a system of equations with only [tex]\( x \)[/tex] as the variable. Let's write down both equations:
[tex]\[ \begin{cases} -x + 3y = 32 \quad \text{(Equation 1)} \\ 7x + 3y = -8 \quad \text{(Equation 2)} \end{cases} \][/tex]
Subtract Equation 1 from Equation 2:
[tex]\[ (7x + 3y) - (-x + 3y) = -8 - 32 \][/tex]
Simplify the resulting equation:
[tex]\[ 7x + 3y + x - 3y = -8 - 32 \][/tex]
[tex]\[ 8x = -40 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = -\frac{40}{8} = -5 \][/tex]
So, [tex]\( x = -5 \)[/tex].
2. Solve for [tex]\( y \)[/tex]:
Now that we have [tex]\( x = -5 \)[/tex], substitute this value into one of the original equations to solve for [tex]\( y \)[/tex]. Let's use Equation 1:
[tex]\[ -x + 3y = 32 \][/tex]
Substitute [tex]\( x = -5 \)[/tex]:
[tex]\[ -(-5) + 3y = 32 \][/tex]
Simplify:
[tex]\[ 5 + 3y = 32 \][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ 3y = 32 - 5 \][/tex]
[tex]\[ 3y = 27 \][/tex]
[tex]\[ y = \frac{27}{3} = 9 \][/tex]
So, [tex]\( y = 9 \)[/tex].
Therefore, the solution to the system of equations is:
[tex]\[ (x, y) = (-5, 9) \][/tex]
[tex]\[ \begin{cases} -x + 3y = 32 \\ 7x + 3y = -8 \end{cases} \][/tex]
we can use the method of elimination to find [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
1. Eliminate variable [tex]\( y \)[/tex]:
To eliminate [tex]\( y \)[/tex], we subtract the first equation from the second equation. This gives us a system of equations with only [tex]\( x \)[/tex] as the variable. Let's write down both equations:
[tex]\[ \begin{cases} -x + 3y = 32 \quad \text{(Equation 1)} \\ 7x + 3y = -8 \quad \text{(Equation 2)} \end{cases} \][/tex]
Subtract Equation 1 from Equation 2:
[tex]\[ (7x + 3y) - (-x + 3y) = -8 - 32 \][/tex]
Simplify the resulting equation:
[tex]\[ 7x + 3y + x - 3y = -8 - 32 \][/tex]
[tex]\[ 8x = -40 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = -\frac{40}{8} = -5 \][/tex]
So, [tex]\( x = -5 \)[/tex].
2. Solve for [tex]\( y \)[/tex]:
Now that we have [tex]\( x = -5 \)[/tex], substitute this value into one of the original equations to solve for [tex]\( y \)[/tex]. Let's use Equation 1:
[tex]\[ -x + 3y = 32 \][/tex]
Substitute [tex]\( x = -5 \)[/tex]:
[tex]\[ -(-5) + 3y = 32 \][/tex]
Simplify:
[tex]\[ 5 + 3y = 32 \][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ 3y = 32 - 5 \][/tex]
[tex]\[ 3y = 27 \][/tex]
[tex]\[ y = \frac{27}{3} = 9 \][/tex]
So, [tex]\( y = 9 \)[/tex].
Therefore, the solution to the system of equations is:
[tex]\[ (x, y) = (-5, 9) \][/tex]