Question

solving systems of equations
solve the following systems of equations by any method. once you solve each system, record the letter that corresponds to the answer in the blank provided. each answer will only be used once. there are blank graphs on the back if you chose to graph the systems.
answers
a: (24, 3) b: (8, 4) c: (12, 17) d: no solution e: (2, 1)
f: (3, 5) g: (8, 2) h: (-6, -1) i: (4, 1) j: infinitely many solutions

1. ( x - 2y = 0 )

( 2x - 5y = -4 )
12 answer: ____

1. ( y = 8 - x )

( 4x - 3y = -3 )
13 answer: ____

1. ( y = x + 5 )

( y = 2x - 7 )
14 answer: ____

1. ( x + y = 3 )

( 2x + 2y = 6 )
15 answer: ____

1. ( x + 3y = 14 )

( -5x + 6y = -28 )
16 answer: ____

1. ( -2x + 6y = 6 )

( y = 2x + 11 )
17 answer: ____

Explanation:

Problem 1: \( x - 2y = 0 \) and \( 2x - 5y = -4 \)

Step 1: Solve first equation for \( x \)

From \( x - 2y = 0 \), we get \( x = 2y \).

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

Substitute into \( 2x - 5y = -4 \): \( 2(2y) - 5y = -4 \)
Simplify: \( 4y - 5y = -4 \) → \( -y = -4 \) → \( y = 4 \).

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

\( x = 2(4) = 8 \). So the solution is \( (8, 4) \), which is option B.

Step 1: Substitute \( y = 8 - x \) into second equation

Substitute into \( 4x - 3y = -3 \): \( 4x - 3(8 - x) = -3 \).

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

Expand: \( 4x - 24 + 3x = -3 \) → \( 7x - 24 = -3 \) → \( 7x = 21 \) → \( x = 3 \).

Step 3: Find \( y \) using \( y = 8 - x \)

\( y = 8 - 3 = 5 \). So the solution is \( (3, 5) \), which is option F.

Step 1: Set equations equal (since both equal \( y \))

\( x + 5 = 2x - 7 \).

Step 2: Solve for \( x \)

Subtract \( x \): \( 5 = x - 7 \) → \( x = 12 \).

Step 3: Find \( y \) using \( y = x + 5 \)

\( y = 12 + 5 = 17 \). So the solution is \( (12, 17) \), which is option C.

Answer:

B. (8, 4)

Problem 2: \( y = 8 - x \) and \( 4x - 3y = -3 \)