Sure, I'll walk you through the detailed, step-by-step solution for the given system of linear equations:
[tex]\[
\left\{
\begin{array}{l}
3x = 7 + y \\
5x - 9y = 41
\end{array}
\right.
\][/tex]
### Step 1: Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]
Starting 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
Now that we have [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], we substitute this expression into the second equation.
The second equation is:
[tex]\[ 5x - 9y = 41 \][/tex]
Substitute [tex]\( y = 3x - 7 \)[/tex]:
[tex]\[ 5x - 9(3x - 7) = 41 \][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]
First, distribute the [tex]\(-9\)[/tex] through the parentheses:
[tex]\[ 5x - 27x + 63 = 41 \][/tex]
Combine like terms:
[tex]\[ -22x + 63 = 41 \][/tex]
Isolate [tex]\( x \)[/tex] by subtracting 63 from both sides:
[tex]\[ -22x = 41 - 63 \][/tex]
[tex]\[ -22x = -22 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 1 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Now that we have [tex]\( x = 1 \)[/tex], substitute this value back into the expression we found for [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 7 \][/tex]
[tex]\[ y = 3(1) - 7 \][/tex]
[tex]\[ y = 3 - 7 \][/tex]
[tex]\[ y = -4 \][/tex]
### Conclusion
So, the solution to the system of equations is:
[tex]\[ x = 1 \][/tex]
[tex]\[ y = -4 \][/tex]
Thus, the solution to the system [tex]\(\{ (3x = 7 + y,\; 5x - 9y = 41) \}\)[/tex] is:
[tex]\[ \left( 1, -4 \right) \][/tex]
