Of course! Let's solve the system of linear equations using the method of linear combination:
Given the system of equations:
[tex]\[
\begin{cases}
-2x + 3y = -15 \\
3x + 2y = -23
\end{cases}
\][/tex]
### Step 1: Align equations
First, let's align our equations one below the other for clarity:
[tex]\[
-2x + 3y = -15 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
3x + 2y = -23 \quad \text{(Equation 2)}
\][/tex]
### Step 2: Eliminate one variable
We aim to eliminate one of the variables by adding or subtracting the equations. To do that, we need to make the coefficients of either [tex]\(x\)[/tex] or [tex]\(y\)[/tex] in both equations the same in absolute value.
Let's eliminate [tex]\(x\)[/tex]. The coefficients of [tex]\(x\)[/tex] in the equations are [tex]\(-2\)[/tex] and [tex]\(3\)[/tex]. The least common multiple (LCM) of 2 and 3 is 6. To make the coefficients of [tex]\(x\)[/tex] in both equations 6, we can multiply the entire first equation by 3 and the entire second equation by 2:
Multiplying Equation 1 by 3:
[tex]\[
3(-2x + 3y) = 3(-15)
\][/tex]
[tex]\[
-6x + 9y = -45 \quad \text{(Equation 3)}
\][/tex]
Multiplying Equation 2 by 2:
[tex]\[
2(3x + 2y) = 2(-23)
\][/tex]
[tex]\[
6x + 4y = -46 \quad \text{(Equation 4)}
\][/tex]
### Step 3: Add the new equations
Now, we add Equation 3 and Equation 4 together to eliminate [tex]\(x\)[/tex]:
[tex]\[
(-6x + 9y) + (6x + 4y) = -45 + (-46)
\][/tex]
[tex]\[
-6x + 6x + 9y + 4y = -91
\][/tex]
[tex]\[
13y = -91
\][/tex]
### Step 4: Solve for [tex]\(y\)[/tex]
To find the value of [tex]\(y\)[/tex], we divide both sides by 13:
[tex]\[
y = \frac{-91}{13}
\][/tex]
[tex]\[
y = -7
\][/tex]
### Step 5: Substitute [tex]\(y\)[/tex] back into one of the original equations
Next, we substitute [tex]\(y = -7\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]. We'll use Equation 1:
[tex]\[
-2x + 3(-7) = -15
\][/tex]
[tex]\[
-2x - 21 = -15
\][/tex]
### Step 6: Solve for [tex]\(x\)[/tex]
We add 21 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
-2x = -15 + 21
\][/tex]
[tex]\[
-2x = 6
\][/tex]
We divide both sides by -2:
[tex]\[
x = \frac{6}{-2}
\][/tex]
[tex]\[
x = -3
\][/tex]
### Final Solution
Therefore, the solution to the system of equations is:
[tex]\[
x = -3, \quad y = -7
\][/tex]
