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Question

Difficulty: MediumSolving Linear Equations

For a certain real number xx, the equation 34(x−3)−13(2x+5)=−2\frac{3}{4}(x - 3) - \frac{1}{3}(2x + 5) = -2 is true. What is the value of 2x−52x - 5?

Answer: 41

Answer

The value of the expression 2x−52x - 5 is 41.
Solving the linear equation by multiplying both sides by the least common denominator of 12 yields the simplified equation 9(x−3)−4(2x+5)=−249(x - 3) - 4(2x + 5) = -24. Expanding the terms gives 9x−27−8x−20=−249x - 27 - 8x - 20 = -24, which simplifies to x−47=−24x - 47 = -24. Adding 47 to both sides gives x=23x = 23. Finally, evaluating the expression 2x−52x - 5 for x=23x = 23 results in 2(23)−5=412(23) - 5 = 41.

Step-by-Step Solution

1
Multiply both sides of the equation by the least common denominator of 12 to eliminate fractions.
9(x−3)−4(2x+5)=−249(x - 3) - 4(2x + 5) = -24
Multiplying by the LCD clears all fractional coefficients, making the equation easier to solve.
2
Distribute the constants and expand the terms on the left side of the equation.
9x−27−8x−20=−249x - 27 - 8x - 20 = -24
Applying the distributive property removes the parentheses.
3
Combine like terms on the left side of the equation.
x−47=−24x - 47 = -24
Simplifying the equation makes it easier to isolate the variable xx.
4
Isolate the variable xx by adding 47 to both sides of the equation.
x=23x = 23
This determines the value of the unknown variable xx.
5
Substitute x=23x = 23 into the target expression 2x−52x - 5.
4141
The question asks for the value of the expression 2x−52x - 5, not just the value of xx.

Key Concept

Solving multi-step linear equations with fractional coefficients by clearing the denominators and then evaluating algebraic expressions.

Alternative Method

Instead of multiplying by the LCD first, distribute the fractions directly: 34x−94−23x−53=−2\frac{3}{4}x - \frac{9}{4} - \frac{2}{3}x - \frac{5}{3} = -2. Combine the xx terms: (34−23)x=112x(\frac{3}{4} - \frac{2}{3})x = \frac{1}{12}x. Combine the constant terms: −94−53=−2712−2012=−4712-\frac{9}{4} - \frac{5}{3} = -\frac{27}{12} - \frac{20}{12} = -\frac{47}{12}. This gives the equation 112x−4712=−2\frac{1}{12}x - \frac{47}{12} = -2. Add 4712\frac{47}{12} to both sides: 112x=−2+4712=−2412+4712=2312\frac{1}{12}x = -2 + \frac{47}{12} = -\frac{24}{12} + \frac{47}{12} = \frac{23}{12}. Multiply by 12 to get x=23x = 23, then evaluate 2x−5=412x - 5 = 41.
Estimated Time:1m 30s
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Topics

Solving Linear Equations

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