Question $\frac{1}{2} x+\frac{1}{6}=\frac{1}{8} x+1$

Studdy Solution

STEP 1

1. The equation $\frac{1}{2} x + \frac{1}{6} = \frac{1}{8} x + 1$ 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 useful.

STEP 2

1. Eliminate fractions by finding a common denominator.
2. Simplify and combine like terms.
3. Isolate the variable $x$ .
4. Check the solution by substituting it back into the original equation.

STEP 3

Identify the least common denominator (LCD) for the fractions involved. The denominators are 2, 6, and 8. The LCD is 24.
Multiply every term in the equation by 24 to eliminate the fractions:
$24 \left(\frac{1}{2} x\right) + 24 \left(\frac{1}{6}\right) = 24 \left(\frac{1}{8} x\right) + 24(1)$

STEP 4

Simplify each term:
$12x + 4 = 3x + 24$

STEP 5

Subtract $3x$ from both sides to start isolating $x$ :
$12x - 3x + 4 = 24$

STEP 6

Combine like terms:
$9x + 4 = 24$

STEP 7

Subtract 4 from both sides to further isolate $x$ :
$9x = 20$

STEP 8

Divide both sides by 9 to solve for $x$ :
$x = \frac{20}{9}$

STEP 9

Check the solution by substituting $x = \frac{20}{9}$ back into the original equation:
Original equation: $\frac{1}{2} x + \frac{1}{6} = \frac{1}{8} x + 1$
Substitute $x = \frac{20}{9}$ :
$\frac{1}{2} \left(\frac{20}{9}\right) + \frac{1}{6} = \frac{1}{8} \left(\frac{20}{9}\right) + 1$

STEP 10

Simplify both sides:
Left side: $\frac{10}{9} + \frac{1}{6} = \frac{10}{9} + \frac{1}{6} = \frac{20}{18} + \frac{3}{18} = \frac{23}{18}$
Right side: $\frac{5}{18} + 1 = \frac{5}{18} + \frac{18}{18} = \frac{23}{18}$
Both sides are equal, confirming the solution is correct.
The solution is:
$\boxed{\frac{20}{9}}$

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