Which of these equations has infinitely many solutions? $\newline$ Choices: $\newline$ (A) $2(x + 3) = 4x + 6$ $\newline$ (B) $3x - 5 + 2x = 5(x - \frac{1}{5})$ $\newline$ (C) $-2x + 3 + 4x = \frac{1}{2}(4x + 6)$ $\newline$ Which statement explains a way you can tell the equation has infinitely many 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 infinitely many solutions?

\newline

Choices:

\newline

(A)

2(x + 3) = 4x + 6

\newline

(B)

3x - 5 + 2x = 5(x - \frac{1}{5})

\newline

(C)

-2x + 3 + 4x = \frac{1}{2}(4x + 6)

\newline

Which statement explains a way you can tell the equation has infinitely many 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

Simplify Equation (A): Simplify equation (A) $2(x + 3) = 4x + 6$ . $\newline$ Calculation: $2x + 6 = 4x + 6$ . $\newline$ Subtract $2x$ from both sides: $6 = 2x + 6$ . $\newline$ Subtract $6$ from both sides: $0 = 2x$ . $\newline$ Divide by $2$ : $x = 0$ .
Check Solution for $x = 0$ : Check if equation (A) simplifies to a true statement for all $x$ . Substitute $x = 0$ back into the original equation: $2(0 + 3) = 4\cdot0 + 6$ . Calculation: $6 = 6$ , which is true. However, this is only true for $x = 0$ , not for all $x$ .
Simplify Equation (B): Simplify equation (B) $3x - 5 + 2x = 5(x - \frac{1}{5})$ . $\newline$ Calculation: $5x - 5 = 5x - 1$ . $\newline$ Subtract $5x$ from both sides: $-5 = -1$ , which is false.
Simplify Equation (C): Simplify equation (C) $-2x + 3 + 4x = \frac{1}{2}(4x + 6)$ . $\newline$ Calculation: $2x + 3 = 2x + 3$ . $\newline$ This simplifies to $3 = 3$ , which is true for all $x$ .