Sure, let's solve the system of equations step-by-step.
Given the system:
[tex]\[
\left\{\begin{array}{l}
3x - y = -11 \\
2x - 3y = -5
\end{array}\right.
\][/tex]
### Step 1: Write the equations in standard form
The system is already in standard form:
[tex]\[
3x - y = -11 \tag{1}
\][/tex]
[tex]\[
2x - 3y = -5 \tag{2}
\][/tex]
### Step 2: Solve one of the equations for one variable
Let's solve equation (1) for [tex]\( y \)[/tex]:
[tex]\[
3x - y = -11
\][/tex]
Add [tex]\( y \)[/tex] to both sides:
[tex]\[
3x = y - 11
\][/tex]
Then add 11 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 3x + 11 \tag{3}
\][/tex]
### Step 3: Substitute [tex]\( y \)[/tex] from equation (3) into equation (2)
Substitute [tex]\( y = 3x + 11 \)[/tex] into [tex]\( 2x - 3y = -5 \)[/tex]:
[tex]\[
2x - 3(3x + 11) = -5
\][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Distribute the [tex]\(-3\)[/tex]:
[tex]\[
2x - 9x - 33 = -5
\][/tex]
Combine like terms:
[tex]\[
-7x - 33 = -5
\][/tex]
Add 33 to both sides:
[tex]\[
-7x = 28
\][/tex]
Divide by [tex]\(-7\)[/tex]:
[tex]\[
x = -4
\][/tex]
### Step 5: Substitute [tex]\( x \)[/tex] back into equation (3) to find [tex]\( y \)[/tex]
Use [tex]\( x = -4 \)[/tex] in equation (3):
[tex]\[
y = 3(-4) + 11
\][/tex]
[tex]\[
y = -12 + 11
\][/tex]
[tex]\[
y = -1
\][/tex]
### Solution
The solution to the system of equations is:
[tex]\[
(x, y) = (-4, -1)
\][/tex]
Hence, there is one solution: [tex]\((-4, -1)\)[/tex].
