Question $\frac{1}{2} x-3=-\frac{1}{6} x+2$

Studdy Solution

STEP 1

1. The equation $\frac{1}{2} x - 3 = -\frac{1}{6} x + 2$ is a linear equation in one variable.
2. The goal is to find the value of $x$ that satisfies the equation.
3. The equation involves fractions, so it may be helpful to eliminate them by finding a common denominator.

STEP 2

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

STEP 3

To eliminate the fractions, find the least common denominator (LCD) of the fractions involved. The denominators are 2 and 6. The LCD is 6.
Multiply every term in the equation by 6 to eliminate the fractions:
$6 \left(\frac{1}{2} x\right) - 6(3) = 6 \left(-\frac{1}{6} x\right) + 6(2)$

STEP 4

Simplify each term:
$3x - 18 = -x + 12$

STEP 5

Combine like terms by adding $x$ to both sides of the equation to get all $x$ terms on one side:
$3x + x - 18 = 12$
Which simplifies to:
$4x - 18 = 12$

STEP 6

Add 18 to both sides to isolate the term with $x$ :
$4x = 12 + 18$
Simplify:
$4x = 30$

STEP 7

Divide both sides by 4 to solve for $x$ :
$x = \frac{30}{4}$
Simplify the fraction:
$x = \frac{15}{2}$

STEP 8

Check the solution by substituting $x = \frac{15}{2}$ back into the original equation:
Original equation:
$\frac{1}{2} \left(\frac{15}{2}\right) - 3 = -\frac{1}{6} \left(\frac{15}{2}\right) + 2$
Simplify each side:
Left side:
$\frac{15}{4} - 3 = \frac{15}{4} - \frac{12}{4} = \frac{3}{4}$
Right side:
$-\frac{15}{12} + 2 = -\frac{5}{4} + \frac{8}{4} = \frac{3}{4}$
Both sides are equal, confirming that the solution is correct.
The solution is:
$\boxed{\frac{15}{2}}$

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