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Which of these equations has infinitely many solutions?\newlineChoices:\newline(A) 2(x+3)=4x+62(x + 3) = 4x + 6\newline(B) 3x−5+2x=5(x−15)3x - 5 + 2x = 5(x - \frac{1}{5})\newline(C) −2x+3+4x=12(4x+6)-2x + 3 + 4x = \frac{1}{2}(4x + 6)\newlineWhich statement explains a way you can tell the equation has infinitely many solutions?\newlineChoices:\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.

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Q. Which of these equations has infinitely many solutions?\newlineChoices:\newline(A) 2(x+3)=4x+62(x + 3) = 4x + 6\newline(B) 3x−5+2x=5(x−15)3x - 5 + 2x = 5(x - \frac{1}{5})\newline(C) −2x+3+4x=12(4x+6)-2x + 3 + 4x = \frac{1}{2}(4x + 6)\newlineWhich statement explains a way you can tell the equation has infinitely many solutions?\newlineChoices:\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.
  1. Simplify Equation (A): Simplify equation (A) 2(x+3)=4x+62(x + 3) = 4x + 6.\newlineCalculation: 2x+6=4x+62x + 6 = 4x + 6.\newlineSubtract 2x2x from both sides: 6=2x+66 = 2x + 6.\newlineSubtract 66 from both sides: 0=2x0 = 2x.\newlineDivide by 22: x=0x = 0.
  2. Check Solution for x=0x = 0: Check if equation (A) simplifies to a true statement for all xx. Substitute x=0x = 0 back into the original equation: 2(0+3)=4⋅0+62(0 + 3) = 4\cdot0 + 6. Calculation: 6=66 = 6, which is true. However, this is only true for x=0x = 0, not for all xx.
  3. Simplify Equation (B): Simplify equation (B) 3x−5+2x=5(x−15)3x - 5 + 2x = 5(x - \frac{1}{5}).\newlineCalculation: 5x−5=5x−15x - 5 = 5x - 1.\newlineSubtract 5x5x from both sides: −5=−1-5 = -1, which is false.
  4. Simplify Equation (C): Simplify equation (C) −2x+3+4x=12(4x+6)-2x + 3 + 4x = \frac{1}{2}(4x + 6).\newlineCalculation: 2x+3=2x+32x + 3 = 2x + 3.\newlineThis simplifies to 3=33 = 3, which is true for all xx.

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