How many solutions does this equation have? 4x - 7/4x + 14 = 7/4(x + 8) + 1/2x Choices: (A)none (B)one (C)two (D)infinitely many

Combine like terms: Simplify the left side of the equation by combining like terms. 4x - 7/4x + 14 = (16/4)x - (7/4)x + 14 = (9/4)x + 14 Simplify right side: Simplify the right side of the equation. 7/4(x + 8) + 1/2x = 7/4x + 14 + 2/4x = 9/4x + 14 Set equal and simplify: Set the simplified left side equal to the simplified right side. (9/4)x + 14 = 9/4x + 14 Isolate constant terms: Subtract (9/4)x from both sides to isolate the constant terms. 14 = 14 Infinite solutions: Since the equation 14 = 14 is always true, the original equation holds for any value of x. Thus, the equation has infinitely many solutions.

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

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Math Problems

Algebra 1

Find the number of solutions to a linear equation

Full solution

Q. How many solutions does this equation have?

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4x - \frac{7}{4}x + 14 = \frac{7}{4}(x + 8) + \frac{1}{2}x

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Choices:

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(A) none

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(B) one

\newline

\newline

(D) infinitely many

Combine like terms: Simplify the left side of the equation by combining like terms. $\newline$ $4x - \frac{7}{4}x + 14$ $\newline$ = $\frac{16}{4}x - \frac{7}{4}x + 14$ $\newline$ = $\frac{9}{4}x + 14$
Simplify right side: Simplify the right side of the equation. $\newline$ $\frac{7}{4}(x + 8) + \frac{1}{2}x$ $\newline$ = $\frac{7}{4}x + 14 + \frac{2}{4}x$ $\newline$ = $\frac{9}{4}x + 14$
Set equal and simplify: Set the simplified left side equal to the simplified right side. $\newline$ $(\frac{9}{4})x + 14 = \frac{9}{4}x + 14$
Isolate constant terms: Subtract $(\frac{9}{4})x$ from both sides to isolate the constant terms. $\newline$ $14 = 14$
Infinite solutions: Since the equation $14 = 14$ is always true, the original equation holds for any value of $x$ . Thus, the equation has infinitely many solutions.