Math  /  Algebra

Questiona 12x13=13(x2)\frac{1}{2} x-\frac{1}{3}=\frac{1}{3}(x-2)

Studdy Solution

STEP 1

1. The given equation is a linear equation in one variable xx.
2. We can use algebraic manipulation to isolate xx on one side of the equation.
3. The operations we use include distributing, combining like terms, and isolating the variable.

STEP 2

1. Distribute the 13\frac{1}{3} on the right-hand side of the equation.
2. Move all terms involving xx to one side of the equation.
3. Move the constant terms to the other side of the equation.
4. Solve for xx by isolating it completely.

STEP 3

Distribute the 13\frac{1}{3} on the right-hand side of the equation.
12x13=13x23 \frac{1}{2} x - \frac{1}{3} = \frac{1}{3} x - \frac{2}{3}

STEP 4

Move all terms involving xx to one side of the equation. Subtract 13x\frac{1}{3} x from both sides.
12x13x13=23 \frac{1}{2} x - \frac{1}{3} x - \frac{1}{3} = - \frac{2}{3}

STEP 5

Combine the xx terms on the left-hand side.
(1213)x13=23 \left( \frac{1}{2} - \frac{1}{3} \right) x - \frac{1}{3} = - \frac{2}{3}
(3626)x13=23 \left( \frac{3}{6} - \frac{2}{6} \right) x - \frac{1}{3} = - \frac{2}{3}
16x13=23 \frac{1}{6} x - \frac{1}{3} = - \frac{2}{3}

STEP 6

Move the constant term 13-\frac{1}{3} to the other side by adding 13\frac{1}{3} to both sides.
16x=23+13 \frac{1}{6} x = - \frac{2}{3} + \frac{1}{3}
16x=13 \frac{1}{6} x = - \frac{1}{3}

STEP 7

Solve for xx by isolating it completely. Multiply both sides by 6 to cancel the fraction on the left-hand side.
x=13×6 x = - \frac{1}{3} \times 6
x=2 x = -2
Solution: x=2 x = -2

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