Question

Which equations have infinitely many solutions? Select TWO apply.
A. 6 ( x - 9 ) = 6x - 54 + x
B. 5 ( x - 8 ) + 10 = 5 ( x + 2 )
C. 20 - 8x - 6 = 2 ( 7 - 4x )
D. 4 ( x + 4 ) = 2 ( 2x+5 ) + 6
E. 2x + 9 = 18 + 2x

Answer 1

To determine which equations have infinitely many solutions, we should look for equations that simplify to a true statement (like \(0 = 0\)) after simplifying both sides. Let's analyze each option:

**A.** \( 6 ( x - 9 ) = 6x - 54 + x \)

Simplifying the left side:
\[ 6x - 54 \]

The right side:
\[ 6x - 54 + x = 7x - 54 \]

So, the equation is:
\[ 6x - 54 = 7x - 54 \]

Subtract \(6x\) from both sides:
\[ -54 = x - 54 \]

Adding \(54\) to both sides:
\[ 0 = x \]
This results in a single solution \( x = 0 \).

**B.** \( 5 ( x - 8 ) + 10 = 5 ( x + 2 ) \)

Left side:
\[ 5x - 40 + 10 = 5x - 30 \]

Right side:
\[ 5x + 10 \]

Setting them equal:
\[ 5x - 30 = 5x + 10 \]

Subtract \(5x\) from both sides:
\[ -30 = 10 \]
This is a false statement, therefore no solutions exist.

**C.** \( 20 - 8x - 6 = 2 ( 7 - 4x ) \)

Left side:
\[ 20 - 6 - 8x = 14 - 8x \]

Right side:
\[ 14 - 8x \]

Setting them equal:
\[ 14 - 8x = 14 - 8x \]

This is a true statement for all values of \(x\), meaning there are infinitely many solutions.

**D.** \( 4 ( x + 4 ) = 2 ( 2x + 5 ) + 6 \)

Left side:
\[ 4x + 16 \]

Right side:
\[ 4x + 10 + 6 = 4x + 16 \]

Setting them equal:
\[ 4x + 16 = 4x + 16 \]

This is also a true statement for all values of \(x\), meaning there are infinitely many solutions.

**E.** \( 2x + 9 = 18 + 2x \)

Subtract \(2x\) from both sides:
\[ 9 = 18 \]
This is a false statement, therefore no solutions exist.

Based on the analysis, the equations that have infinitely many solutions are:
**C.** \(20 - 8x - 6 = 2 (7 - 4x)\)
**D.** \(4 (x + 4) = 2 (2x + 5) + 6\)