Question8) $\frac{1}{2} x+3=\frac{1}{4} x+5$

Studdy Solution

STEP 1

1. The equation $\frac{1}{2} x + 3 = \frac{1}{4} x + 5$ is a linear equation.
2. We need to solve for the variable $x$ .
3. The equation involves fractions, so we may need to eliminate them for easier manipulation.

STEP 2

1. Eliminate fractions by finding a common denominator.
2. Isolate the variable $x$ on one side of the equation.
3. Solve for $x$ .
4. Check the solution by substituting it back into the original equation.

STEP 3

Identify the least common denominator (LCD) for the fractions $\frac{1}{2}$ and $\frac{1}{4}$ . The LCD is 4.
Multiply every term in the equation by 4 to eliminate the fractions:
$4 \left(\frac{1}{2} x\right) + 4(3) = 4 \left(\frac{1}{4} x\right) + 4(5)$

STEP 4

Simplify each term:
$2x + 12 = x + 20$

STEP 5

To isolate $x$ , subtract $x$ from both sides of the equation:
$2x - x + 12 = 20$

STEP 6

Simplify the equation:
$x + 12 = 20$

STEP 7

Subtract 12 from both sides to solve for $x$ :
$x = 20 - 12$

STEP 8

Simplify the right-hand side:
$x = 8$

STEP 9

Check the solution by substituting $x = 8$ back into the original equation:
$\frac{1}{2}(8) + 3 = \frac{1}{4}(8) + 5$
Calculate each side:
Left side: $\frac{1}{2}(8) + 3 = 4 + 3 = 7$
Right side: $\frac{1}{4}(8) + 5 = 2 + 5 = 7$
Both sides are equal, confirming the solution is correct.
The solution is:
$\boxed{8}$

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