Question

6.5a · solving systems of equations by substitution
solve each system of equation by substitution.

1. ( x = y + 7 )

( x + y = 9 )

1. ( 2x - y = 4 )

( y = 2x - 4 )

1. ( x = -y + 5 )

( x - 4y = 10 )

1. ( y = -6x + 4 )

( 2x + 2y = -12 )

1. ( 5x - 3y = -11 )

( x = 2y + 2 )

1. ( y = 3x - 5 )

( 4x - y = 3 )

Explanation:

Problem 1:

Step 1: Substitute \( x = y + 7 \) into \( x + y = 9 \)

Substitute \( x \) in the second equation: \( (y + 7) + y = 9 \)

Step 2: Solve for \( y \)

Simplify: \( 2y + 7 = 9 \)
Subtract 7: \( 2y = 9 - 7 = 2 \)
Divide by 2: \( y = \frac{2}{2} = 1 \)

Step 3: Solve for \( x \)

Substitute \( y = 1 \) into \( x = y + 7 \): \( x = 1 + 7 = 8 \)

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

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

Step 2: Simplify

Simplify: \( 2x - 2x + 4 = 4 \) → \( 4 = 4 \)
This is an identity, so there are infinitely many solutions (the lines are coincident).

Step 1: Substitute \( x = -y + 5 \) into \( x - 4y = 10 \)

Substitute \( x \): \( (-y + 5) - 4y = 10 \)

Step 2: Solve for \( y \)

Simplify: \( -5y + 5 = 10 \)
Subtract 5: \( -5y = 10 - 5 = 5 \)
Divide by -5: \( y = \frac{5}{-5} = -1 \)

Step 3: Solve for \( x \)

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

Answer:

\( x = 8, y = 1 \)

Problem 2: