Question

solve the following systems using any method.

1. $2x - y = 2$

$4x + 3y = 24$

1. $x = y + 1$

$2x - 2y = -5$

1. $y = 2x - 1$

$y = x + 1$

1. $6x + 21y = 132$

$6x - 4y = 32$

Explanation:

Problem 45:

Step 1: Solve the first equation for \( y \)

From \( 2x - y = 2 \), we can rearrange to get \( y = 2x - 2 \).

Step 2: Substitute \( y = 2x - 2 \) into the second equation

Substitute into \( 4x + 3y = 24 \):
\( 4x + 3(2x - 2) = 24 \)
Expand the equation: \( 4x + 6x - 6 = 24 \)
Combine like terms: \( 10x - 6 = 24 \)
Add 6 to both sides: \( 10x = 30 \)
Divide by 10: \( x = 3 \)

Step 3: Find \( y \) using \( y = 2x - 2 \)

Substitute \( x = 3 \): \( y = 2(3) - 2 = 6 - 2 = 4 \)

Step 1: Substitute \( x = y + 1 \) into the second equation

Substitute into \( 2x - 2y = -5 \):
\( 2(y + 1) - 2y = -5 \)
Expand: \( 2y + 2 - 2y = -5 \)
Simplify: \( 2 = -5 \)
This is a contradiction, so there is no solution.

Step 1: Set the two equations equal to each other

Since \( y = 2x - 1 \) and \( y = x + 1 \), we have \( 2x - 1 = x + 1 \)

Step 2: Solve for \( x \)

Subtract \( x \) from both sides: \( x - 1 = 1 \)
Add 1 to both sides: \( x = 2 \)

Step 3: Find \( y \) using \( y = x + 1 \)

Substitute \( x = 2 \): \( y = 2 + 1 = 3 \)

Answer:

\( x = 3 \), \( y = 4 \)

Problem 46: