Question $\frac{x}{2}-1=\frac{2}{3} x+5$

Studdy Solution

STEP 1

1. The equation $\frac{x}{2} - 1 = \frac{2}{3}x + 5$ is a linear equation.
2. We need to solve for the variable $x$ .
3. The equation involves fractions, so we may need to find a common denominator.

STEP 2

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

STEP 3

To eliminate fractions, find a common denominator for the terms involving $x$ . The denominators are 2 and 3, so the common denominator is 6.
Multiply every term in the equation by 6 to eliminate the fractions:
$6 \left( \frac{x}{2} \right) - 6(1) = 6 \left( \frac{2}{3}x \right) + 6(5)$

STEP 4

Simplify each term:
$3x - 6 = 4x + 30$

STEP 5

Move all terms involving $x$ to one side of the equation and constant terms to the other side. Subtract $4x$ from both sides:
$3x - 4x - 6 = 30$

STEP 6

Simplify the equation:
$-x - 6 = 30$

STEP 7

Add 6 to both sides to isolate the term with $x$ :
$-x = 36$

STEP 8

Multiply both sides by $-1$ to solve for $x$ :
$x = -36$

STEP 9

Check the solution by substituting $x = -36$ back into the original equation:
$\frac{-36}{2} - 1 = \frac{2}{3}(-36) + 5$
Calculate each side separately:
Left side: $\frac{-36}{2} - 1 = -18 - 1 = -19$
Right side: $\frac{2}{3}(-36) + 5 = -24 + 5 = -19$
Both sides are equal, confirming the solution is correct.
The solution is:
$\boxed{-36}$

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