Which of these equations has no solutions? $\newline$ Choices: $\newline$ (A) $4(x + 3) = 4(x - 3)$ $\newline$ (B) $3x + x - 3 = \frac{1}{3}(12x - 9)$ $\newline$ (C) $2x + 3 - 4x = 4x + 3$ $\newline$ Which statement explains a way you can tell the equation has no solutions? $\newline$ Choices: $\newline$ (A) It is equivalent to an equation that has the same variable terms but different constant terms on each side of the equal sign. $\newline$ (B) It is equivalent to an equation that has the same variable terms and the same constant terms on each side of the equal sign. $\newline$ (C) It is equivalent to an equation that has different variable terms on each side of the equation.

Full solution

Q. Which of these equations has no solutions?

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Choices:

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(A)

4(x + 3) = 4(x - 3)

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(B)

3x + x - 3 = \frac{1}{3}(12x - 9)

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(C)

2x + 3 - 4x = 4x + 3

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Which statement explains a way you can tell the equation has no solutions?

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Choices:

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(A) It is equivalent to an equation that has the same variable terms but different constant terms on each side of the equal sign.

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(B) It is equivalent to an equation that has the same variable terms and the same constant terms on each side of the equal sign.

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Distribute and Simplify: Simplify equation (A) $4(x + 3) = 4(x - 3)$ . Distribute the $4$ on both sides: $4x + 12 = 4x - 12$ . Subtract $4x$ from both sides: $12 = -12$ .
No Solution for Equation (A): Since $12$ does not equal $-12$ , equation (A) simplifies to a false statement, indicating no solutions.
Combine and Distribute: Simplify equation (B) $3x + x - 3 = \frac{1}{3}(12x - 9)$ . Combine like terms on the left: $4x - 3$ . Distribute the $\frac{1}{3}$ on the right: $4x - 3$ . Both sides are equal, suggesting infinite solutions.
Solve for x in Equation (C): Simplify equation (C) $2x + 3 - 4x = 4x + 3$ . Combine like terms on the left: $-2x + 3$ . The equation becomes $-2x + 3 = 4x + 3$ . Subtract $3$ from both sides: $-2x = 4x$ . Add $2x$ to both sides: $0 = 6x$ . Divide by $6$ : $x = 0$ . This equation has a solution.