To solve the system of equations:
[tex]\[
\begin{array}{c}
2x - y = -19 \\
-3x + 5y = 39
\end{array}
\][/tex]
we can follow these steps:
### 1. Write down the system of equations:
[tex]\[
\begin{array}{c}
2x - y = -19 \quad \text{(Equation 1)} \\
-3x + 5y = 39 \quad \text{(Equation 2)}
\end{array}
\][/tex]
### 2. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] from Equation 1:
Start with Equation 1:
[tex]\[ 2x - y = -19 \][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 19 \][/tex]
### 3. Substitute [tex]\( y \)[/tex] into Equation 2:
Now take the expression for [tex]\( y \)[/tex] from Equation 1 and substitute it into Equation 2:
[tex]\[-3x + 5(2x + 19) = 39\][/tex]
### 4. Simplify and solve for [tex]\( x \)[/tex]:
Expand the terms in the equation:
[tex]\[-3x + 10x + 95 = 39\][/tex]
Combine like terms:
[tex]\[ 7x + 95 = 39 \][/tex]
Subtract 95 from both sides:
[tex]\[ 7x = 39 - 95 \][/tex]
[tex]\[ 7x = -56 \][/tex]
Divide by 7:
[tex]\[ x = -8 \][/tex]
### 5. Substitute [tex]\( x \)[/tex] back into the expression for [tex]\( y \)[/tex]:
Substitute [tex]\( x = -8 \)[/tex] into [tex]\( y = 2x + 19 \)[/tex]:
[tex]\[ y = 2(-8) + 19 \][/tex]
[tex]\[ y = -16 + 19 \][/tex]
[tex]\[ y = 3 \][/tex]
### 6. State the solution:
The solution to the system of equations is:
[tex]\[ x = -8 \][/tex]
[tex]\[ y = 3 \][/tex]
So, the values that satisfy both equations are [tex]\( x = -8 \)[/tex] and [tex]\( y = 3 \)[/tex].
