To solve the given system of linear equations:
[tex]\[
\begin{cases}
3x + 2y = 2 \\
x - 2y = 6
\end{cases}
\][/tex]
we can use the method of substitution or elimination. Here, we'll use the elimination method for a detailed, step-by-step solution:
### Step 1: Align the equations
Let's rewrite the system of equations for clarity:
[tex]\[
3x + 2y = 2 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
x - 2y = 6 \quad \text{(Equation 2)}
\][/tex]
### Step 2: Eliminate one of the variables
To eliminate [tex]\( y \)[/tex], we can add Equation 1 and Equation 2. Notice that the coefficients of [tex]\( y \)[/tex] in these two equations are [tex]\( +2 \)[/tex] and [tex]\( -2 \)[/tex], respectively. Adding these equations will cancel [tex]\( y \)[/tex] out.
[tex]\[
(3x + 2y) + (x - 2y) = 2 + 6
\][/tex]
### Step 3: Simplify the resulting equation
When we add the equations:
[tex]\[
3x + 2y + x - 2y = 8
\][/tex]
The [tex]\( +2y \)[/tex] and [tex]\( -2y \)[/tex] terms cancel each other out, resulting in:
[tex]\[
4x = 8
\][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
To find [tex]\( x \)[/tex], divide both sides of the equation by 4:
[tex]\[
x = \frac{8}{4} = 2
\][/tex]
### Step 5: Substitute [tex]\( x \)[/tex] back into one of the original equations
We now have [tex]\( x = 2 \)[/tex]. Substitute this value into Equation 2 to determine [tex]\( y \)[/tex]:
[tex]\[
x - 2y = 6
\][/tex]
[tex]\[
2 - 2y = 6
\][/tex]
### Step 6: Solve for [tex]\( y \)[/tex]
Isolate [tex]\( y \)[/tex] by first subtracting 2 from both sides:
[tex]\[
-2y = 6 - 2
\][/tex]
[tex]\[
-2y = 4
\][/tex]
Then divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
y = \frac{4}{-2} = -2
\][/tex]
### Step 7: Verify the solution
Check the values [tex]\( x = 2 \)[/tex] and [tex]\( y = -2 \)[/tex] in both original equations to ensure they satisfy both:
For Equation 1:
[tex]\[
3(2) + 2(-2) = 6 - 4 = 2 \quad \text{(True)}
\][/tex]
For Equation 2:
[tex]\[
2 - 2(-2) = 2 + 4 = 6 \quad \text{(True)}
\][/tex]
Both equations are satisfied, so the solution to the system of equations is:
[tex]\[
\boxed{(2, -2)}
\][/tex]
