Question

Which equations have infinitely many solutions? Select TWO that 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 need to simplify each equation and see if they result in a true statement or a contradiction.

Let's evaluate each option:

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

Simplifying both sides:
Left Side: \(6x - 54\)
Right Side: \(6x - 54 + x\) simplifies to \(7x - 54\).

The equation becomes: \(6x - 54 = 7x - 54\)

Subtracting \(6x\) from both sides:
\(-54 = x - 54\)

Adding 54 to both sides gives \(x = 0\).
This does not have infinitely many solutions.

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

Simplifying both sides:
Left Side: \(5x - 40 + 10 = 5x - 30\)
Right Side: \(5x + 10\).

Now the equation becomes \(5x - 30 = 5x + 10\).

Subtracting \(5x\) from both sides yields:
\(-30 = 10\), which is a contradiction.
Thus, this does not have infinitely many solutions.

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

Simplifying both sides:
Left Side: \(20 - 6 - 8x = 14 - 8x\).
Right Side: This simplifies to \(14 - 8x\).

Thus, the equation becomes:
\(14 - 8x = 14 - 8x\).

This is true for all values of \(x\), indicating infinitely many solutions.

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

Simplifying both sides:
Left Side: \(4x + 16\),
Right Side: \(4x + 10 + 6 = 4x + 16\).

Thus, the equation becomes:
\(4x + 16 = 4x + 16\).

This is true for all values of \(x\), indicating infinitely many solutions.

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

Subtracting \(2x\) from both sides, we have:
\(9 = 18\), which is a contradiction. Thus, this does not have infinitely many solutions.

Based on this analysis, the equations that have infinitely many solutions are:

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

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