Let's solve the given system of linear equations step by step.
The system of equations provided is:
[tex]\[
\left\{\begin{array}{l}
x - 3y = 7 \\
2x + 4 = 21
\end{array}\right.
\][/tex]
### Step 1: Solve the second equation for [tex]\(x\)[/tex]
First, we start with the second equation:
[tex]\[
2x + 4 = 21
\][/tex]
Subtract 4 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
2x = 21 - 4
\][/tex]
Simplify the right-hand side:
[tex]\[
2x = 17
\][/tex]
Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{17}{2}
\][/tex]
Thus, the value of [tex]\(x\)[/tex] is:
[tex]\[
x = 8.5
\][/tex]
### Step 2: Substitute the value of [tex]\(x\)[/tex] into the first equation to solve for [tex]\(y\)[/tex]
Now, we substitute [tex]\(x = 8.5\)[/tex] into the first equation:
[tex]\[
x - 3y = 7
\][/tex]
Replace [tex]\(x\)[/tex] with 8.5:
[tex]\[
8.5 - 3y = 7
\][/tex]
Subtract 8.5 from both sides to isolate the term with [tex]\(y\)[/tex]:
[tex]\[
-3y = 7 - 8.5
\][/tex]
Simplify the right-hand side:
[tex]\[
-3y = -1.5
\][/tex]
Divide both sides by -3 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-1.5}{-3}
\][/tex]
Simplify the division:
[tex]\[
y = 0.5
\][/tex]
### Step 3: Confirm the solution
The solution to the system of equations:
[tex]\[
x = 8.5
\][/tex]
[tex]\[
y = 0.5
\][/tex]
### Verification
To verify, we substitute [tex]\(x = 8.5\)[/tex] and [tex]\(y = 0.5\)[/tex] back into the original equations:
First equation:
[tex]\[
8.5 - 3(0.5) = 7
\][/tex]
[tex]\[
8.5 - 1.5 = 7
\][/tex]
[tex]\[
7 = 7 \][/tex]
This is true.
Second equation:
[tex]\[
2(8.5) + 4 = 21
\][/tex]
[tex]\[
17 + 4 = 21
\][/tex]
[tex]\[
21 = 21 \][/tex]
This is also true.
Therefore, the solution [tex]\(x = 8.5\)[/tex] and [tex]\(y = 0.5\)[/tex] satisfies both equations.
