Question $\frac{1}{3} x+\frac{1}{2}=\frac{1}{4} x+1$

Studdy Solution

STEP 1

1. The equation $\frac{1}{3} x + \frac{1}{2} = \frac{1}{4} x + 1$ is a linear equation.
2. The goal is to solve for the variable $x$ .
3. The equation involves fractions, so finding a common denominator may be necessary.

STEP 2

1. Eliminate fractions by finding a common denominator.
2. Move all terms involving $x$ to one side of the equation.
3. Move constant terms to the opposite side.
4. Solve for $x$ .
5. Check the solution by substituting it back into the original equation.

STEP 3

Identify a common denominator for the fractions. The denominators are 3, 2, and 4. The least common multiple is 12.
Multiply every term in the equation by 12 to eliminate the fractions:
$12 \left(\frac{1}{3} x\right) + 12 \left(\frac{1}{2}\right) = 12 \left(\frac{1}{4} x\right) + 12(1)$

STEP 4

Simplify each term:
$4x + 6 = 3x + 12$

STEP 5

Move all terms involving $x$ to one side by subtracting $3x$ from both sides:
$4x - 3x + 6 = 12$

STEP 6

Simplify the equation:
$x + 6 = 12$

STEP 7

Move the constant term to the opposite side by subtracting 6 from both sides:
$x = 12 - 6$

STEP 8

Simplify the equation:
$x = 6$

STEP 9

Check the solution by substituting $x = 6$ back into the original equation:
$\frac{1}{3}(6) + \frac{1}{2} = \frac{1}{4}(6) + 1$

STEP 10

Simplify both sides:
$2 + \frac{1}{2} = \frac{3}{2} + 1$
Convert to a common denominator:
$\frac{5}{2} = \frac{5}{2}$
Both sides are equal, confirming the solution is correct.
The solution is:
$\boxed{6}$

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