Question

1. $y = 6x - 4$

$y = -4x - 4$

1. $3x - 4y = 7$

$y = 5x - 6$

1. $y = -8x - 4$

$-16x - 2y = 1$

1. $-7x + 6y = 25$

$6x - 6y = -24$

Explanation:

Let's solve each system of equations one by one.

Problem 4: \( y = 6x - 4 \) and \( y = -4x - 4 \)

Step 1: Set the two equations equal to each other

Since both equations are solved for \( y \), we can set \( 6x - 4 = -4x - 4 \)

Step 2: Solve for \( x \)

Add \( 4x \) to both sides: \( 6x + 4x - 4 = -4 \)
Simplify: \( 10x - 4 = -4 \)
Add 4 to both sides: \( 10x = 0 \)
Divide by 10: \( x = 0 \)

Step 3: Substitute \( x = 0 \) into one of the equations to find \( y \)

Using \( y = 6x - 4 \): \( y = 6(0) - 4 = -4 \)

Step 1: Substitute \( y = 5x - 6 \) into \( 3x - 4y = 7 \)

\( 3x - 4(5x - 6) = 7 \)

Step 2: Simplify and solve for \( x \)

Distribute: \( 3x - 20x + 24 = 7 \)
Combine like terms: \( -17x + 24 = 7 \)
Subtract 24: \( -17x = -17 \)
Divide by -17: \( x = 1 \)

Step 3: Substitute \( x = 1 \) into \( y = 5x - 6 \)

\( y = 5(1) - 6 = -1 \)

Step 1: Substitute \( y = -8x - 4 \) into \( -16x - 2y = 1 \)

\( -16x - 2(-8x - 4) = 1 \)

Step 2: Simplify and solve for \( x \)

Distribute: \( -16x + 16x + 8 = 1 \)
Simplify: \( 8 = 1 \)
This is a contradiction, so there is no solution.

Answer:

The solution is \( x = 0 \), \( y = -4 \)

Problem 6: \( 3x - 4y = 7 \) and \( y = 5x - 6 \)