Math Snap

STEP 1

1. The equation $\frac{1}{6}(x-1) + x = \frac{1}{2}$ is a linear equation.
2. The goal is to solve for the variable $x$.
3. Basic algebraic operations such as distribution, combining like terms, and solving linear equations are required.

STEP 2

1. Distribute the fraction $\frac{1}{6}$ across the terms inside the parentheses.
2. Combine like terms.
3. Isolate the variable $x$.
4. Verify the solution by substituting it back into the original equation.

STEP 3

Distribute $\frac{1}{6}$ across the terms inside the parentheses:
$$\frac{1}{6}(x-1) = \frac{1}{6}x - \frac{1}{6}$$ The equation becomes:
$$\frac{1}{6}x - \frac{1}{6} + x = \frac{1}{2}$$

STEP 4

Combine like terms. Notice that $\frac{1}{6}x$ and $x$ are like terms:
$$\frac{1}{6}x + x = \frac{1}{6}x + \frac{6}{6}x = \frac{7}{6}x$$ The equation now is:
$$\frac{7}{6}x - \frac{1}{6} = \frac{1}{2}$$

STEP 5

Isolate the variable $x$ by first eliminating the constant term $-\frac{1}{6}$ from the left side. Add $\frac{1}{6}$ to both sides:
$$\frac{7}{6}x = \frac{1}{2} + \frac{1}{6}$$ To add the fractions, find a common denominator:
$$\frac{1}{2} = \frac{3}{6}$$ Thus:
$$\frac{7}{6}x = \frac{3}{6} + \frac{1}{6} = \frac{4}{6}$$ Simplify $\frac{4}{6}$ to $\frac{2}{3}$:
$$\frac{7}{6}x = \frac{2}{3}$$ Now, solve for $x$ by multiplying both sides by the reciprocal of $\frac{7}{6}$, which is $\frac{6}{7}$:
$$x = \frac{2}{3} \times \frac{6}{7} = \frac{12}{21}$$ Simplify $\frac{12}{21}$ to $\frac{4}{7}$:
$$x = \frac{4}{7}$$

Verify the solution by substituting $x = \frac{4}{7}$ back into the original equation:
Original equation:
$$\frac{1}{6}(x-1) + x = \frac{1}{2}$$ Substitute $x = \frac{4}{7}$:
$$\frac{1}{6}\left(\frac{4}{7} - 1\right) + \frac{4}{7}$$ Calculate $\frac{4}{7} - 1$:
$$\frac{4}{7} - \frac{7}{7} = -\frac{3}{7}$$ Substitute back:
$$\frac{1}{6}\left(-\frac{3}{7}\right) + \frac{4}{7} = -\frac{3}{42} + \frac{4}{7}$$ Convert $\frac{4}{7}$ to have a denominator of 42:
$$\frac{4}{7} = \frac{24}{42}$$ Add the fractions:
$$-\frac{3}{42} + \frac{24}{42} = \frac{21}{42}$$ Simplify $\frac{21}{42}$ to $\frac{1}{2}$:
$$\frac{1}{2} = \frac{1}{2}$$ The left side equals the right side, confirming the solution is correct.
The solution is:
$$x = \frac{4}{7}$$