Which of these equations has no solutions? $\newline$ Choices: $\newline$ (A) $\frac{1}{3}(9x + 3) = 4x + 5 + x$ $\newline$ (B) $-3(x + 5) = -x - 2x - 15$ $\newline$ (C) $-2x + 5 + 8x = 3(2x - 1)$ $\newline$ 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 either 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 either side of the equal sign. $\newline$ (C) It is equivalent to an equation that has different variable terms on either side of the equation.

Full solution

Q. Which of these equations has no solutions?

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Choices:

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(A)

\frac{1}{3}(9x + 3) = 4x + 5 + x

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(B)

-3(x + 5) = -x - 2x - 15

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(C)

-2x + 5 + 8x = 3(2x - 1)

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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 either 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 either side of the equal sign.

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Simplify equation (A): Simplify equation (A) $\frac{1}{3}(9x + 3) = 4x + 5 + x$ . Distribute and combine like terms: $\newline$ $\frac{1}{3} \times 9x + \frac{1}{3} \times 3 = 4x + 5 + x$ $\newline$ $3x + 1 = 5x + 5$
Solve for x: Solve for x in equation (A): $\newline$ $3x + 1 = 5x + 5$ $\newline$ $1 - 5 = 5x - 3x$ $\newline$ $-4 = 2x$ $\newline$ $x = -2$
Simplify equation (B): Simplify equation (B) $-3(x + 5) = -x - 2x - 15$ . Distribute and combine like terms: $\newline$ $-3x - 15 = -3x - 15$
Infinitely many solutions: Since equation (B) simplifies to $-3x - 15 = -3x - 15$ , which is always true, it has infinitely many solutions.
Simplify equation (C): Simplify equation (C) $-2x + 5 + 8x = 3(2x - 1)$ . Distribute and combine like terms: $6x + 5 = 6x - 3$
Solve for x: Solve for x in equation (C): $\newline$ $6x + 5 = 6x - 3$ $\newline$ $5 = -3$