Question

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

Answer 1

To determine which equations have infinitely many solutions, we need to simplify each equation and see if they lead to an identity (e.g., something that simplifies to a true statement for all values of \( x \)) or if they become a contradiction (e.g., something that leads to a false statement).

Let's evaluate each option:

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

Simplifying the left side:
\[
6(x - 9) = 6x - 54
\]
The right side simplifies to:
\[
6x - 54 + x = 7x - 54
\]

So the equation becomes:
\[
6x - 54 = 7x - 54
\]
Subtract \( 6x \) from both sides:
\[
-54 = x - 54
\]
Adding \( 54 \) to both sides gives:
\[
0 = x
\]
This equation has a unique solution (not infinitely many).

---

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

Simplify the left side:
\[
5(x - 8) + 10 = 5x - 40 + 10 = 5x - 30
\]

The right side is:
\[
5(x + 2) = 5x + 10
\]

Thus, the equation simplifies to:
\[
5x - 30 = 5x + 10
\]
Now, subtract \( 5x \) from both sides:
\[
-30 = 10
\]
This is a contradiction, which means this equation has no solutions (not infinitely many).

---

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

On the left:
\[
20 - 8x - 6 = 14 - 8x
\]

On the right:
\[
2(7 - 4x) = 14 - 8x
\]

Thus, we have:
\[
14 - 8x = 14 - 8x
\]
This is an identity, so this equation has infinitely many solutions.

---

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

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

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

So the equation simplifies to:
\[
4x + 16 = 4x + 16
\]
This is another identity, so this equation also has infinitely many solutions.

---

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

Subtracting \( 2x \) from both sides:
\[
9 = 18
\]
This is a contradiction, which means this equation has no solutions.

From the analysis, the two equations that have infinitely many solutions are:

**C. \( 20 - 8x - 6 = 2(7 - 4x) \)**
**D. \( 4(x + 4) = 2(2x + 5) + 6 \)**