Sure, let's solve this system of equations step-by-step:
We start with the given system of equations:
[tex]\[
\left\{
\begin{array}{l}
2(2x + 7) - 3x = 5 \\
4x + 3(x + 2y) = 11
\end{array}
\right.
\][/tex]
Step 1: Simplify the first equation
Let's simplify the first equation:
[tex]\[
2(2x + 7) - 3x = 5
\][/tex]
Distribute the 2:
[tex]\[
4x + 14 - 3x = 5
\][/tex]
Combine like terms:
[tex]\[
x + 14 = 5
\][/tex]
Isolate [tex]\(x\)[/tex] by subtracting 14 from both sides:
[tex]\[
x = 5 - 14
\][/tex]
Simplify:
[tex]\[
x = -9
\][/tex]
Step 2: Substitute [tex]\(x = -9\)[/tex] into the second equation
Now, we substitute [tex]\(x = -9\)[/tex] into the second equation to solve for [tex]\(y\)[/tex]:
[tex]\[
4(-9) + 3((-9) + 2y) = 11
\][/tex]
Simplify inside the parentheses:
[tex]\[
-36 + 3(-9 + 2y) = 11
\][/tex]
Distribute the 3:
[tex]\[
-36 + 3(-9) + 6y = 11
\][/tex]
Simplify:
[tex]\[
-36 - 27 + 6y = 11
\][/tex]
Combine like terms:
[tex]\[
-63 + 6y = 11
\][/tex]
Isolate [tex]\(6y\)[/tex] by adding 63 to both sides:
[tex]\[
6y = 74
\][/tex]
Solve for [tex]\(y\)[/tex] by dividing both sides by 6:
[tex]\[
y = \frac{74}{6}
\][/tex]
Simplify:
[tex]\[
y = \frac{37}{3}
\][/tex]
Step 3: Verify the solution
We have found [tex]\(x = -9\)[/tex] and [tex]\(y = \frac{37}{3}\)[/tex]. To verify, we substitute these values back into the original simultaneous equations to check if they hold true.
First equation:
[tex]\[
2(2x + 7) - 3x = 5
\][/tex]
[tex]\[
2(2(-9) + 7) - 3(-9) = 5
\][/tex]
[tex]\[
2(-18 + 7) + 27 = 5
\][/tex]
[tex]\[
2(-11) + 27 = 5
\][/tex]
[tex]\[
-22 + 27 = 5
\][/tex]
[tex]\[
5 = 5 \quad \text{(True)}
\][/tex]
Second equation:
[tex]\[
4x + 3(x + 2y) = 11
\][/tex]
[tex]\[
4(-9) + 3((-9) + 2(\frac{37}{3})) = 11
\][/tex]
[tex]\[
-36 + 3(-9 + \frac{74}{3}) = 11
\][/tex]
[tex]\[
-36 + 3(\frac{-27 + 74}{3}) = 11
\][/tex]
[tex]\[
-36 + 3(\frac{47}{3}) = 11
\][/tex]
[tex]\[
-36 + 47 = 11
\][/tex]
[tex]\[
11 = 11 \quad \text{(True)}
\][/tex]
Both equations hold true, so our solution is correct.
Thus, the solution to the system is:
[tex]\[
x = -9, \quad y = \frac{37}{3}
\][/tex]
