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Difficulty: HardSolving Linear Equations

If xx satisfies the equation 3(x−2)5−2x−13=115\frac{3(x - 2)}{5} - \frac{2x - 1}{3} = \frac{1}{15}, what is the value of the expression 2x+72x + 7?

Answer: -21

Answer

The final value of the expression is −21-21.
Solving the equation by first multiplying both sides by the least common denominator of 15 yields the simplified equation 9(x−2)−5(2x−1)=19(x - 2) - 5(2x - 1) = 1. Distributing and combining terms yields −x−13=1-x - 13 = 1, which gives x=−14x = -14. Substituting this value into the expression 2x+72x + 7 yields −21-21.

Step-by-Step Solution

1
Multiply the entire equation by the least common multiple of the denominators (1515) to eliminate all fractions.
9(x−2)−5(2x−1)=19(x - 2) - 5(2x - 1) = 1
Multiplying by the least common multiple of 5 and 3 eliminates the fractions and simplifies the equation.
2
Distribute the coefficients (99 and −5-5) to their respective terms inside the parentheses.
9x−18−10x+5=19x - 18 - 10x + 5 = 1
Distributive property allows us to remove parentheses. Note that distributing −5-5 to −1-1 results in +5+5.
3
Combine the variable terms (9x9x and −10x-10x) and the constant terms (−18-18 and 55) on the left side of the equation.
−x−13=1-x - 13 = 1
Combining like terms simplifies the expression to prepare for isolating the variable.
4
Isolate the variable term −x-x by adding 1313 to both sides, then solve for xx by multiplying by −1-1.
x=−14x = -14
Adding 1313 yields −x=14-x = 14, and multiplying by −1-1 isolates xx to find its value.
5
Substitute x=−14x = -14 into the given expression 2x+72x + 7.
2(−14)+7=−212(-14) + 7 = -21
The question asks for the value of the expression 2x+72x + 7, not just xx itself.

Key Concept

Solving multi-step linear equations involving fractions, distributive property with negative signs, and evaluating algebraic expressions.

Alternative Method

Instead of multiplying by the least common multiple first, you can separate the fractions: 35x−65−23x+13=115\frac{3}{5}x - \frac{6}{5} - \frac{2}{3}x + \frac{1}{3} = \frac{1}{15}. Combining the xx terms gives (915−1015)x=−115x(\frac{9}{15} - \frac{10}{15})x = -\frac{1}{15}x. Combining the constant terms gives −1815+515=−1315-\frac{18}{15} + \frac{5}{15} = -\frac{13}{15}. The equation becomes −115x−1315=115-\frac{1}{15}x - \frac{13}{15} = \frac{1}{15}. Multiplying the entire equation by 1515 yields −x−13=1-x - 13 = 1, which gives x=−14x = -14, and substituting into 2x+72x + 7 yields −21-21.
Estimated Time:2m 0s
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Topics

Solving Linear Equations

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