Which equations have infinitely many solutions? Select TWO that apply.

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

GPT-4o mini · Answer

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. $6 (x - 9) = 6 x - 54 + x$

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

The equation becomes: $6 x - 54 = 7 x - 54$

Subtracting $6 x$ 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: $5 x - 40 + 10 = 5 x - 30$ Right Side: $5 x + 10$ .

Now the equation becomes $5 x - 30 = 5 x + 10$ .

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

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

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

Thus, the equation becomes: $14 - 8 x = 14 - 8 x$ .

This is true for all values of $x$ , indicating infinitely many solutions.

D. $4 (x + 4) = 2 (2 x + 5) + 6$

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

Thus, the equation becomes: $4 x + 16 = 4 x + 16$ .

This is true for all values of $x$ , indicating infinitely many solutions.

E. $2 x + 9 = 18 + 2 x$

Subtracting $2 x$ 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. $20 - 8 x - 6 = 2 (7 - 4 x)$

D. $4 (x + 4) = 2 (2 x + 5) + 6$