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How many solutions does this equation have?\newline4x−74x+14=74(x+8)+12x4x - \frac{7}{4}x + 14 = \frac{7}{4}(x + 8) + \frac{1}{2}x\newlineChoices:\newline(A) none\newline(B) one\newline(C) two\newline(D) infinitely many

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Find the number of solutions to a linear equationright chevron icon

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Q. How many solutions does this equation have?\newline4x−74x+14=74(x+8)+12x4x - \frac{7}{4}x + 14 = \frac{7}{4}(x + 8) + \frac{1}{2}x\newlineChoices:\newline(A) none\newline(B) one\newline(C) two\newline(D) infinitely many
  1. Combine like terms: Simplify the left side of the equation by combining like terms. \newline4x−74x+144x - \frac{7}{4}x + 14 \newline= 164x−74x+14\frac{16}{4}x - \frac{7}{4}x + 14 \newline= 94x+14\frac{9}{4}x + 14
  2. Simplify right side: Simplify the right side of the equation. \newline74(x+8)+12x\frac{7}{4}(x + 8) + \frac{1}{2}x \newline= 74x+14+24x\frac{7}{4}x + 14 + \frac{2}{4}x \newline= 94x+14\frac{9}{4}x + 14
  3. Set equal and simplify: Set the simplified left side equal to the simplified right side.\newline(94)x+14=94x+14(\frac{9}{4})x + 14 = \frac{9}{4}x + 14
  4. Isolate constant terms: Subtract (94)x(\frac{9}{4})x from both sides to isolate the constant terms.\newline14=1414 = 14
  5. Infinite solutions: Since the equation 14=1414 = 14 is always true, the original equation holds for any value of xx. Thus, the equation has infinitely many solutions.

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