Solve the system of equations: 4x - y = 24 and 3y - 2 = 4 - (x - y)

This is a system of two linear equations with two variables. We can solve it using substitution or elimination.

Step 1: Simplify the second equation.

The second equation is: \( 3y - 2 = 4 - (x - y) \)

Distribute the negative sign: \( 3y - 2 = 4 - x + y \)

Move all terms to one side to set the equation to 0 or to isolate a variable:

Add \( x \) to both sides: \( x + 3y - 2 = 4 + y \)

Subtract \( y \) from both sides: \( x + 2y - 2 = 4 \)

Add 2 to both sides: \( x + 2y = 6 \)

Step 2: Rewrite the system of equations.

Now the system is:

1. \( 4x - y = 24 \)
2. \( x + 2y = 6 \)

Step 3: Solve the system using substitution or elimination.

Let's use the substitution method. From the second equation, we can express \( x \) in terms of \( y \):

\( x = 6 - 2y \)

Step 4: Substitute this expression for \( x \) into the first equation.

The first equation is: \( 4x - y = 24 \)

Substitute \( x = 6 - 2y \): \( 4(6 - 2y) - y = 24 \)

Step 5: Solve for \( y \).

Distribute the 4: \( 24 - 8y - y = 24 \)

Combine like terms: \( 24 - 9y = 24 \)

Subtract 24 from both sides: \( -9y = 0 \)

Divide by -9: \( y = 0 \)

Step 6: Substitute the value of \( y \) back into the equation for \( x \).

\( x = 6 - 2y \)

Substitute \( y = 0 \): \( x = 6 - 2(0) \)

\( x = 6 - 0 \)

\( x = 6 \)

Step 7: Check the solution.

Substitute \( x = 6 \) and \( y = 0 \) into both original equations:

First equation: \( 4x - y = 24 \) \( \Rightarrow \) \( 4(6) - 0 = 24 \) \( \Rightarrow \) \( 24 = 24 \) (True)

Second equation: \( 3y - 2 = 4 - (x - y) \) \( \Rightarrow \) \( 3(0) - 2 = 4 - (6 - 0) \) \( \Rightarrow \) \( -2 = 4 - 6 \) \( \Rightarrow \) \( -2 = -2 \) (True)

Answer: x = 6, y = 0