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Math  /  Algebra

QuestionSolve for xx : 16(x−1)+x=12x=□\begin{array}{l} \frac{1}{6}(x-1)+x=\frac{1}{2} \\ x=\square \end{array}

Studdy Solution

STEP 1

1. The equation 16(x−1)+x=12\frac{1}{6}(x-1) + x = \frac{1}{2} is a linear equation.
2. The goal is to solve for the variable x x .
3. Basic algebraic operations such as distribution, combining like terms, and solving linear equations are required.

STEP 2

1. Distribute the fraction 16\frac{1}{6} across the terms inside the parentheses.
2. Combine like terms.
3. Isolate the variable x x .
4. Verify the solution by substituting it back into the original equation.

STEP 3

Distribute 16\frac{1}{6} across the terms inside the parentheses:
16(x−1)=16x−16 \frac{1}{6}(x-1) = \frac{1}{6}x - \frac{1}{6}
The equation becomes:
16x−16+x=12 \frac{1}{6}x - \frac{1}{6} + x = \frac{1}{2}

STEP 4

Combine like terms. Notice that 16x\frac{1}{6}x and xx are like terms:
16x+x=16x+66x=76x \frac{1}{6}x + x = \frac{1}{6}x + \frac{6}{6}x = \frac{7}{6}x
The equation now is:
76x−16=12 \frac{7}{6}x - \frac{1}{6} = \frac{1}{2}

STEP 5

Isolate the variable x x by first eliminating the constant term −16-\frac{1}{6} from the left side. Add 16\frac{1}{6} to both sides:
76x=12+16 \frac{7}{6}x = \frac{1}{2} + \frac{1}{6}
To add the fractions, find a common denominator:
12=36 \frac{1}{2} = \frac{3}{6}
Thus:
76x=36+16=46 \frac{7}{6}x = \frac{3}{6} + \frac{1}{6} = \frac{4}{6}
Simplify 46\frac{4}{6} to 23\frac{2}{3}:
76x=23 \frac{7}{6}x = \frac{2}{3}
Now, solve for x x by multiplying both sides by the reciprocal of 76\frac{7}{6}, which is 67\frac{6}{7}:
x=23×67=1221 x = \frac{2}{3} \times \frac{6}{7} = \frac{12}{21}
Simplify 1221\frac{12}{21} to 47\frac{4}{7}:
x=47 x = \frac{4}{7}

STEP 6

Verify the solution by substituting x=47 x = \frac{4}{7} back into the original equation:
Original equation:
16(x−1)+x=12 \frac{1}{6}(x-1) + x = \frac{1}{2}
Substitute x=47 x = \frac{4}{7} :
16(47−1)+47 \frac{1}{6}\left(\frac{4}{7} - 1\right) + \frac{4}{7}
Calculate 47−1\frac{4}{7} - 1:
47−77=−37 \frac{4}{7} - \frac{7}{7} = -\frac{3}{7}
Substitute back:
16(−37)+47=−342+47 \frac{1}{6}\left(-\frac{3}{7}\right) + \frac{4}{7} = -\frac{3}{42} + \frac{4}{7}
Convert 47\frac{4}{7} to have a denominator of 42:
47=2442 \frac{4}{7} = \frac{24}{42}
Add the fractions:
−342+2442=2142 -\frac{3}{42} + \frac{24}{42} = \frac{21}{42}
Simplify 2142\frac{21}{42} to 12\frac{1}{2}:
12=12 \frac{1}{2} = \frac{1}{2}
The left side equals the right side, confirming the solution is correct.
The solution is:
x=47 x = \frac{4}{7}

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