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Difficulty: MediumSolving Linear Equations

A laboratory technician mixes two solutions. The volume of the first solution is represented by 12(x−5)\frac{1}{2}(x - 5) liters, and the volume of the second solution is represented by 13(2x+1)\frac{1}{3}(2x + 1) liters, where xx is a positive real number. If the sum of the volumes of these two solutions is 66 liters, what is the value of 3x−43x - 4?

  1. A
    66
  2. B
    99
  3. 1717Answer
  4. D
    3030
  5. E
    5353

Answer

The value of the expression is 1717.
The sum of the volumes of the two solutions is 66 liters, which translates to the linear equation 12(x−5)+13(2x+1)=6\frac{1}{2}(x - 5) + \frac{1}{3}(2x + 1) = 6. Multiplying the entire equation by the least common multiple of the denominators, 66, yields 3(x−5)+2(2x+1)=363(x - 5) + 2(2x + 1) = 36. Distributing the constants leads to 3x−15+4x+2=363x - 15 + 4x + 2 = 36. Combining like terms gives 7x−13=367x - 13 = 36. Adding 1313 to both sides results in 7x=497x = 49, which gives x=7x = 7. Substituting x=7x = 7 into the expression 3x−43x - 4 yields 3(7)−4=173(7) - 4 = 17, which is the correct value.

Step-by-Step Solution

1
Set up the linear equation based on the word problem context.
12(x−5)+13(2x+1)=6\frac{1}{2}(x - 5) + \frac{1}{3}(2x + 1) = 6
The sum of the volumes of the two solutions is given as 66 liters.
2
Clear the fractions by multiplying both sides of the equation by the least common multiple of the denominators.
3(x−5)+2(2x+1)=363(x - 5) + 2(2x + 1) = 36
Multiplying by 66 eliminates the fractions and simplifies the equation for solving.
3
Distribute the coefficients to eliminate the parentheses.
3x−15+4x+2=363x - 15 + 4x + 2 = 36
Applying the distributive property allows like terms to be grouped.
4
Combine like terms and isolate the variable xx.
7x−13=36  ⟹  7x=49  ⟹  x=77x - 13 = 36 \implies 7x = 49 \implies x = 7
Adding 1313 to both sides and dividing by 77 solves for the value of the variable xx.
5
Evaluate the expression requested in the question using the solved value of xx.
3(7)−4=21−4=173(7) - 4 = 21 - 4 = 17
Substituting x=7x = 7 into 3x−43x - 4 yields the final requested value.

Key Concept

Solving linear equations with fractional coefficients by clearing denominators and evaluating variable expressions.

Alternative Method

Instead of clearing the fractions immediately, we can distribute the fractions first: 12x−2.5+23x+13=6\frac{1}{2}x - 2.5 + \frac{2}{3}x + \frac{1}{3} = 6. Converting to fractions with a common denominator of 6, we get 36x−156+46x+26=6\frac{3}{6}x - \frac{15}{6} + \frac{4}{6}x + \frac{2}{6} = 6, which simplifies to 76x−136=6\frac{7}{6}x - \frac{13}{6} = 6. Adding 136\frac{13}{6} to both sides yields 76x=496\frac{7}{6}x = \frac{49}{6}, so 7x=497x = 49, and x=7x = 7. Substituting x=7x = 7 into the expression 3x−43x - 4 yields 1717.
Estimated Time:1m 30s
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Topics

Solving Linear Equations

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