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Math  /  Algebra

Question12x−13=14x−15\frac{1}{2} x-\frac{1}{3}=\frac{1}{4} x-\frac{1}{5}

Studdy Solution

STEP 1

1. The equation involves fractions and linear terms.
2. We need to isolate the variable x x by performing operations to eliminate the fractions.

STEP 2

1. Eliminate the fractions by finding a common denominator.
2. Simplify the equation.
3. Isolate the variable x x .

STEP 3

First, identify the least common denominator (LCD) for all fractions in the equation. The denominators are 2, 3, 4, and 5. The LCD of these numbers is 60.

STEP 4

Multiply every term in the equation by the LCD (60) to eliminate the fractions:
60(12x)−60(13)=60(14x)−60(15) 60 \left(\frac{1}{2} x\right) - 60 \left(\frac{1}{3}\right) = 60 \left(\frac{1}{4} x\right) - 60 \left(\frac{1}{5}\right)
This simplifies to:
30x−20=15x−12 30x - 20 = 15x - 12

STEP 5

Rearrange the equation to get all terms involving x x on one side and constant terms on the other side. Subtract 15x 15x from both sides:
30x−15x−20=−12 30x - 15x - 20 = -12
This simplifies to:
15x−20=−12 15x - 20 = -12
Add 20 to both sides to isolate the term with x x :
15x=−12+20 15x = -12 + 20
15x=8 15x = 8

STEP 6

Solve for x x by dividing both sides by 15:
x=815 x = \frac{8}{15}
The value of x x is:
815 \boxed{\frac{8}{15}}

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