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Difficulty: Very hardLinear Equations in One Variable

If xx is the solution to the equation 14(23x−8)−56(12−35x)=13(x−9)−1\frac{1}{4}\left(\frac{2}{3}x - 8\right) - \frac{5}{6}\left(12 - \frac{3}{5}x\right) = \frac{1}{3}(x - 9) - 1, what is the value of 12x−5\frac{1}{2}x - 5?

  1. 7Answer
  2. B
    -11
  3. C
    1
  4. D
    -29

Answer

7
The correct answer is obtained by first distributing the fractions across the terms in the parentheses, which yields 16x−2−10+12x=13x−4\frac{1}{6}x - 2 - 10 + \frac{1}{2}x = \frac{1}{3}x - 4. Combining the like terms on the left side gives 23x−12=13x−4\frac{2}{3}x - 12 = \frac{1}{3}x - 4. Subtracting 13x\frac{1}{3}x from both sides and adding 1212 to both sides yields 13x=8\frac{1}{3}x = 8, which means x=24x = 24. Substituting 2424 for xx in the expression 12x−5\frac{1}{2}x - 5 gives 12(24)−5=12−5=7\frac{1}{2}(24) - 5 = 12 - 5 = 7.

Step-by-Step Solution

1
Distribute the coefficients outside the parentheses on both sides of the equation.
16x−2−10+12x=13x−3−1\frac{1}{6}x - 2 - 10 + \frac{1}{2}x = \frac{1}{3}x - 3 - 1
To eliminate parentheses and simplify the terms.
2
Combine the like terms on the left and right sides of the equation.
23x−12=13x−4\frac{2}{3}x - 12 = \frac{1}{3}x - 4
To group the coefficients of xx and the constant values.
3
Isolate the variable xx by subtracting 13x\frac{1}{3}x and adding 1212 to both sides of the equation.
13x=8  ⟹  x=24\frac{1}{3}x = 8 \implies x = 24
To solve for xx.
4
Substitute the value of xx into the expression 12x−5\frac{1}{2}x - 5 to find the final value.
12(24)−5=12−5=7\frac{1}{2}(24) - 5 = 12 - 5 = 7
To evaluate the requested expression.

Key Concept

Solving multi-step linear equations in one variable involving distribution, fractions, and grouping like terms.

Alternative Method

Instead of distributing fractions first, we can multiply the entire equation by the least common multiple of all denominators, which is 1212. This eliminates all fractions immediately: 3(23x−8)−10(12−35x)=4(x−9)−123\left(\frac{2}{3}x - 8\right) - 10\left(12 - \frac{3}{5}x\right) = 4(x - 9) - 12. Expanding this gives 2x−24−120+6x=4x−36−12  ⟹  8x−144=4x−48  ⟹  4x=96  ⟹  x=242x - 24 - 120 + 6x = 4x - 36 - 12 \implies 8x - 144 = 4x - 48 \implies 4x = 96 \implies x = 24. Substituting x=24x = 24 into the expression gives 77.
Estimated Time:3m 0s
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Topics

Linear Equations in One Variable

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