How many solutions does this equation have? 5/6(x - 3) + 1/3x = 1/2x + 2/3x - 5 Choices: (A)none (B)one (C)two (D)infinitely many

Combine and Simplify Left Side: Simplify the left side of the equation by combining terms. 5/6(x - 3) + 1/3x = 5/6x - 5/2 + 1/3x = (10/6x + 2/6x) - 5/2 = 12/6x - 5/2 = 2x - 5/2 Combine and Simplify Right Side: Simplify the right side of the equation by combining terms. 1/2x + 2/3x - 5 = (3/6x + 4/6x) - 5 = 7/6x - 5 Set Equal and Solve for x: Set the simplified left side equal to the simplified right side and solve for x. 2x - 5/2 = 7/6x - 5 Multiply all terms by 6 to clear the fractions: 12x - 15 = 7x - 30 Subtract 7x from both sides: 5x - 15 = -30 Add 15 to both sides: 5x = -15 Divide by 5: x = -3

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How many solutions does this equation have? $\newline$ $\frac{5}{6}(x - 3) + \frac{1}{3}x = \frac{1}{2}x + \frac{2}{3}x - 5$ $\newline$ 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?

\newline

\frac{5}{6}(x - 3) + \frac{1}{3}x = \frac{1}{2}x + \frac{2}{3}x - 5

\newline

Choices:

\newline

(A) none

\newline

(B) one

\newline

\newline

(D) infinitely many

Combine and Simplify Left Side: Simplify the left side of the equation by combining terms. $\newline$ $\frac{5}{6}(x - 3) + \frac{1}{3}x$ $\newline$ = $\frac{5}{6}x - \frac{5}{2} + \frac{1}{3}x$ $\newline$ = $(\frac{10}{6}x + \frac{2}{6}x) - \frac{5}{2}$ $\newline$ = $\frac{12}{6}x - \frac{5}{2}$ $\newline$ = $2x - \frac{5}{2}$
Combine and Simplify Right Side: Simplify the right side of the equation by combining terms. $\newline$ $\frac{1}{2}x + \frac{2}{3}x - 5$ $\newline$ = $\left(\frac{3}{6}x + \frac{4}{6}x\right) - 5$ $\newline$ = $\frac{7}{6}x - 5$
Set Equal and Solve for x: Set the simplified left side equal to the simplified right side and solve for $x$ .
$2x - \frac{5}{2} = \frac{7}{6}x - 5$
Multiply all terms by $6$ to clear the fractions:
$12x - 15 = 7x - 30$
Subtract $7x$ from both sides:
$5x - 15 = -30$
Add $15$ to both sides:
$5x = -15$
Divide by $5$ :
$x = -3$