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Difficulty: MediumLinear Equations in One Variable

If xx satisfies the linear equation 3(x−1)2−x+54=2x−33\frac{3(x - 1)}{2} - \frac{x + 5}{4} = \frac{2x - 3}{3}, what is the value of 4x+74x + 7?

  1. A
    15
  2. 19Answer
  3. C
    23
  4. D
    27
  5. E
    31

Answer

19
Multiplying the entire equation by the least common denominator 12 yields 18(x−1)−3(x+5)=4(2x−3)18(x - 1) - 3(x + 5) = 4(2x - 3). Expanding the terms gives 18x−18−3x−15=8x−1218x - 18 - 3x - 15 = 8x - 12, which simplifies to 15x−33=8x−1215x - 33 = 8x - 12. Subtracting 8x8x and adding 3333 gives 7x=217x = 21, so x=3x = 3. Substituting x=3x = 3 into 4x+74x + 7 produces 4(3)+7=194(3) + 7 = 19.

Step-by-Step Solution

1
Find the least common denominator (LCD) for all fractions in the equation.
The LCD of 2, 4, and 3 is 12.
Clearing denominators simplifies the linear equation to integer coefficients.
2
Multiply both sides of the equation by 12.
12 \cdot \left(\frac{3(x - 1)}{2}\right) - 12 \cdot \left(\frac{x + 5}{4}\right) = 12 \cdot \left(\frac{2x - 3}{3}\right) \implies 6 \cdot 3(x - 1) - 3(x + 5) = 4(2x - 3)
Multiplying each term by 12 eliminates all fractions.
3
Expand the terms on both sides of the equation.
18(x - 1) - 3(x + 5) = 8x - 12 \implies 18x - 18 - 3x - 15 = 8x - 12
Distribute the coefficients across each set of parentheses.
4
Combine like terms and solve for xx.
15x - 33 = 8x - 12 \implies 7x = 21 \implies x = 3
Isolate the variable term xx on one side of the equation.
5
Evaluate the requested expression 4x+74x + 7 using x=3x = 3.
4(3) + 7 = 12 + 7 = 19
Substitute the value of xx into 4x+74x + 7 to find the final requested value.

Key Concept

Solving multi-step linear equations in one variable with fractional coefficients by clearing denominators.
Estimated Time:1m 30s
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Topics

Linear Equations in One Variable

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