Sure, let's solve the system of linear equations step-by-step:
[tex]\[
\left\{\begin{array}{l}
3x + 5y = -14 \\
x - y = 6
\end{array}\right.
\][/tex]
### Step 1: Solving for [tex]\(x\)[/tex] from the second equation
First, we'll isolate [tex]\(x\)[/tex] in the second equation:
[tex]\[
x - y = 6
\][/tex]
Rearrange to solve for [tex]\(x\)[/tex]:
[tex]\[
x = y + 6
\][/tex]
### Step 2: Substituting [tex]\(x\)[/tex] into the first equation
Next, we'll substitute [tex]\(x = y + 6\)[/tex] into the first equation:
[tex]\[
3x + 5y = -14
\][/tex]
Substitute [tex]\(x\)[/tex]:
[tex]\[
3(y + 6) + 5y = -14
\][/tex]
### Step 3: Simplifying and solving for [tex]\(y\)[/tex]
Distribute the 3:
[tex]\[
3y + 18 + 5y = -14
\][/tex]
Combine like terms:
[tex]\[
8y + 18 = -14
\][/tex]
Subtract 18 from both sides:
[tex]\[
8y = -14 - 18
\][/tex]
[tex]\[
8y = -32
\][/tex]
Divide both sides by 8:
[tex]\[
y = -4
\][/tex]
### Step 4: Solving for [tex]\(x\)[/tex]
Now that we have [tex]\(y = -4\)[/tex], we can find [tex]\(x\)[/tex] using the equation [tex]\(x = y + 6\)[/tex]:
[tex]\[
x = -4 + 6
\][/tex]
[tex]\[
x = 2
\][/tex]
### Solution
The solution to the system of equations is:
[tex]\[
x = 2, \quad y = -4
\][/tex]
Thus, the system of equations:
[tex]\[
\left\{\begin{array}{l}
3 x + 5 y = -14 \\
x - y = 6
\end{array}\right.
\][/tex]
is satisfied by [tex]\(x = 2\)[/tex] and [tex]\(y = -4\)[/tex].
