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






















































B.

C.

D.

Answer 1

To determine which equations have infinitely many solutions, we should simplify each equation and check if they hold true for all values of \(x\).

Let's evaluate each equation one by one:

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

Left side:
\(6(x - 9) = 6x - 54\)

Right side:
\(6x - 54 + x = 7x - 54\)

Setting both sides equal:
\(6x - 54 = 7x - 54\)

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

So,
\(0 = x\)

This equation has a unique solution (\(x = 0\)), not infinitely many solutions.

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

Left side:
\(5(x - 8) + 10 = 5x - 40 + 10 = 5x - 30\)

Right side:
\(5(x + 2) = 5x + 10\)

Setting both sides equal:
\(5x - 30 = 5x + 10\)

Subtracting \(5x\) from both sides:
\(-30 = 10\)

This is a contradiction, which means there are no solutions.

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

Left side:
\(20 - 8x - 6 = 14 - 8x\)

Right side:
\(2(7 - 4x) = 14 - 8x\)

Setting both sides equal:
\(14 - 8x = 14 - 8x\)

This equation is always true for all \(x\). Thus, it has infinitely many solutions.

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

Left side:
\(4(x + 4) = 4x + 16\)

Right side:
\(2(2x + 5) + 6 = 4x + 10 + 6 = 4x + 16\)

Setting both sides equal:
\(4x + 16 = 4x + 16\)

This equation is also always true for all \(x\). Thus, it has infinitely many solutions.

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In conclusion, the equations that have infinitely many solutions are:

**C.** and **D.**