Of course, let's go through the solution step by step:
We are given the system of linear equations:
[tex]\[
\left\{
\begin{array}{l}
x - 3y = 7 \\
2x + y = 21
\end{array}
\right.
\][/tex]
Step 1: Solve one of the equations for one variable
First, we solve the second equation for [tex]\( y \)[/tex]:
[tex]\[
2x + y = 21
\][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[
y = 21 - 2x
\][/tex]
Step 2: Substitute this expression into the other equation
Next, we substitute [tex]\( y = 21 - 2x \)[/tex] into the first equation:
[tex]\[
x - 3(21 - 2x) = 7
\][/tex]
Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[
x - 63 + 6x = 7
\][/tex]
Combine like terms:
[tex]\[
7x - 63 = 7
\][/tex]
Add 63 to both sides:
[tex]\[
7x = 70
\][/tex]
Divide by 7:
[tex]\[
x = 10
\][/tex]
Step 3: Substitute [tex]\( x \)[/tex] back into the expression for [tex]\( y \)[/tex]
Substitute [tex]\( x = 10 \)[/tex] back into the expression [tex]\( y = 21 - 2x \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[
y = 21 - 2(10)
\][/tex]
Calculate:
[tex]\[
y = 21 - 20
\][/tex]
[tex]\[
y = 1
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
x = 10, \quad y = 1
\][/tex]
So, the solution is [tex]\((x, y) = (10, 1)\)[/tex].
