Asked by jrjty

Kate begins solving the equation StartFraction 2 Over 3 EndFraction left-parenthesis 6 x minus 3 right-parenthesis equals StartFraction one-half EndFraction left-parenthesis 6 x minus 4 left-parenthesis.(6x – 3) = StartFraction 2 Over 3 EndFraction left-parenthesis 6 x minus 3 right-parenthesis equals StartFraction one-half EndFraction left-parenthesis 6 x minus 4 left-parenthesis.(6x – 4). Her work is correct and is shown below.

StartFraction 2 Over 3 EndFraction left-parenthesis 6 x minus 3 right-parenthesis equals StartFraction one-half EndFraction left-parenthesis 6 x minus 4 left-parenthesis.(6x – 3) = StartFraction 2 Over 3 EndFraction left-parenthesis 6 x minus 3 right-parenthesis equals StartFraction one-half EndFraction left-parenthesis 6 x minus 4 left-parenthesis.(6x – 4)
4x – 2 = 3x – 2
When she adds 2 to both sides, the equation 4x = 3x results. Which solution will best illustrate what happens to x ?

The equation has infinite solutions.
The equation has one solution: x = 0.
The equation has one solution: x = StartFraction 4 Over 3 EndFraction..

All Answers 1

Answered by GPT-5 mini AI
Start with (2/3)(6x − 3) = (1/2)(6x − 4).

Expand both sides:
(2/3)·6x − (2/3)·3 = 4x − 2
(1/2)·6x − (1/2)·4 = 3x − 2

So 4x − 2 = 3x − 2. Add 2 to both sides: 4x = 3x. Subtract 3x: x = 0.

Check: (2/3)(6·0 − 3) = (2/3)(−3) = −2 and (1/2)(6·0 − 4) = (1/2)(−4) = −2. Both sides match.

Answer: The equation has one solution: x = 0.