To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the given system of equations:
[tex]\[
\begin{aligned}
\frac{3x - y}{y - 2} &= 4 \\
\frac{x + 2y}{2x - y} &= 3
\end{aligned}
\][/tex]
let's solve them step by step.
### Step 1: Solve the first equation
First, isolate the fraction in the first equation:
[tex]\[
\frac{3x - y}{y - 2} = 4
\][/tex]
Multiply both sides by [tex]\( y - 2 \)[/tex] to eliminate the denominator:
[tex]\[
3x - y = 4(y - 2)
\][/tex]
Distribute the 4 on the right side:
[tex]\[
3x - y = 4y - 8
\][/tex]
Rearrange to gather all [tex]\( y \)[/tex]-terms on one side:
[tex]\[
3x - y - 4y = -8
\][/tex]
Combine like terms:
[tex]\[
3x - 5y = -8 \quad \text{(1)}
\][/tex]
### Step 2: Solve the second equation
Next, isolate the fraction in the second equation:
[tex]\[
\frac{x + 2y}{2x - y} = 3
\][/tex]
Multiply both sides by [tex]\( 2x - y \)[/tex]:
[tex]\[
x + 2y = 3(2x - y)
\][/tex]
Distribute the 3 on the right side:
[tex]\[
x + 2y = 6x - 3y
\][/tex]
Rearrange to gather all [tex]\( y \)[/tex]-terms on one side:
[tex]\[
x - 6x + 2y + 3y = 0
\][/tex]
Combine like terms:
[tex]\[
-5x + 5y = 0
\][/tex]
Divide both sides by 5:
[tex]\[
-x + y = 0 \quad \text{(2)}
\][/tex]
### Step 3: Solve the linear system
Now we solve the linear system composed of equations (1) and (2):
[tex]\[
\begin{aligned}
3x - 5y &= -8 \quad \text{(1)} \\
-x + y &= 0 \quad \text{(2)}
\end{aligned}
\][/tex]
From equation (2), we can express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = x
\][/tex]
Substitute [tex]\( y = x \)[/tex] into equation (1):
[tex]\[
3x - 5(x) = -8
\][/tex]
Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[
3x - 5x = -8 \\
-2x = -8 \\
x = 4
\][/tex]
Since [tex]\( y = x \)[/tex], we have:
[tex]\[
y = 4
\][/tex]
### Conclusion
The solution to the system of equations is:
[tex]\[
x = 4, \quad y = 4
\][/tex]
