Sure, let's solve the given system of equations step by step. The system of equations is:
[tex]\[
\begin{cases}
3x = 7 + y \\
5x - 9y = 41
\end{cases}
\][/tex]
### Step 1: Solve for [tex]\( y \)[/tex] from the first equation.
Start with the first equation:
[tex]\[ 3x = 7 + y \][/tex]
Rearrange it to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 7 \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] into the second equation.
Take the expression for [tex]\( y \)[/tex] from the first equation and substitute it into the second equation [tex]\( 5x - 9y = 41 \)[/tex]:
[tex]\[ 5x - 9(3x - 7) = 41 \][/tex]
Now, distribute [tex]\(-9\)[/tex] through the parentheses:
[tex]\[ 5x - 27x + 63 = 41 \][/tex]
Combine like terms:
[tex]\[ -22x + 63 = 41 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex].
Isolate [tex]\( x \)[/tex] by moving 63 to the right-hand side:
[tex]\[ -22x = 41 - 63 \][/tex]
[tex]\[ -22x = -22 \][/tex]
Divide both sides by [tex]\(-22\)[/tex]:
[tex]\[ x = 1 \][/tex]
### Step 4: Find [tex]\( y \)[/tex] using the value of [tex]\( x \)[/tex].
Now that we have [tex]\( x = 1 \)[/tex], substitute it back into the expression for [tex]\( y \)[/tex] that we derived from the first equation:
[tex]\[ y = 3(1) - 7 \][/tex]
[tex]\[ y = 3 - 7 \][/tex]
[tex]\[ y = -4 \][/tex]
So the solution to the system of equations is:
[tex]\[ x = 1 \][/tex]
[tex]\[ y = -4 \][/tex]
Therefore, the solution set for the system of equations is:
[tex]\[ (x, y) = (1, -4) \][/tex]
