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Difficulty: MediumSolving One-Variable Linear Equations

Find the value of xx that satisfies the equation:

x−34+2x+13=5x6 \frac{x - 3}{4} + \frac{2x + 1}{3} = \frac{5x}{6}

Enter the exact value of xx.

Answer: 5

Answer

5
By finding the least common denominator (12) and multiplying every term by it, the fractions are cleanly eliminated. After applying the distributive property and combining like terms (11x−5=10x11x - 5 = 10x), isolating the variable xx yields exactly 5.

Step-by-Step Solution

1
Multiply all terms by the least common denominator (12).
12(x−34)+12(2x+13)=12(5x6)12 \left(\frac{x - 3}{4}\right) + 12 \left(\frac{2x + 1}{3}\right) = 12 \left(\frac{5x}{6}\right)
Multiplying by the LCD eliminates all fractions from the equation, simplifying further steps.
2
Simplify the resulting fractions.
3(x−3)+4(2x+1)=2(5x)3(x - 3) + 4(2x + 1) = 2(5x)
Dividing 12 by each respective denominator leaves an integer multiplier for the numerators.
3
Distribute the multipliers across the terms in the parentheses.
3x−9+8x+4=10x3x - 9 + 8x + 4 = 10x
The distributive property is required to remove the parentheses.
4
Combine like terms on the left side of the equation.
11x−5=10x11x - 5 = 10x
Grouping xx terms (3x+8x3x + 8x) and constant terms (−9+4-9 + 4) simplifies the expression.
5
Isolate the variable xx.
x=5x = 5
Subtracting 10x10x from both sides gives x−5=0x - 5 = 0. Adding 55 to both sides results in x=5x = 5.

Key Concept

Solving linear equations involving rational expressions by multiplying by a least common denominator.
Estimated Time:1m 30s
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Topics

Solving One-Variable Linear Equations

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