To solve the given system of equations:
[tex]\[
\begin{cases}
\frac{3x - y}{y - 2} = 4 \\
\frac{x + 2y}{2x - y} = 3
\end{cases}
\][/tex]
we can follow these detailed steps:
### Step 1: Simplify the equations
1. Simplify Equation 1:
[tex]\[
\frac{3x - y}{y - 2} = 4
\][/tex]
Multiply both sides by [tex]\(y - 2\)[/tex]:
[tex]\[
3x - y = 4(y - 2)
\][/tex]
Distribute the 4 on the right-hand side:
[tex]\[
3x - y = 4y - 8
\][/tex]
Bring all terms involving [tex]\(y\)[/tex] to one side:
[tex]\[
3x - y - 4y = -8
\][/tex]
Combine like terms:
[tex]\[
3x - 5y = -8
\][/tex]
2. Simplify Equation 2:
[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-hand side:
[tex]\[
x + 2y = 6x - 3y
\][/tex]
Bring all terms involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex] to one side:
[tex]\[
x + 2y - 6x + 3y = 0
\][/tex]
Combine like terms:
[tex]\[
-5x + 5y = 0
\][/tex]
Divide both sides by -5:
[tex]\[
x - y = 0 \quad \text{or} \quad x = y
\][/tex]
### Step 2: Substitute [tex]\( x = y \)[/tex] into the first simplified equation
Substitute [tex]\( x = y \)[/tex] into [tex]\(3x - 5y = -8\)[/tex]:
[tex]\[
3(y) - 5(y) = -8
\][/tex]
Which simplifies to:
[tex]\[
3y - 5y = -8
\][/tex]
Combine like terms:
[tex]\[
-2y = -8
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
y = 4
\][/tex]
### Step 3: Find [tex]\( x \)[/tex] using [tex]\( x = y \)[/tex]
Using [tex]\( x = y \)[/tex]:
[tex]\[
x = 4
\][/tex]
### Step 4: Present the solution
The solution to the system of equations is:
[tex]\[
x = 4, \quad y = 4
\][/tex]
So, the coordinates [tex]\( (x, y) \)[/tex] that satisfy both equations are:
[tex]\[
(4, 4)
\][/tex]
