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Difficulty: HardSolving One-Variable Linear Equations

An algebraic relationship is given by the equation:

35(10k−25)−14(8−16k)=12(14k−10)−2(k−9) \frac{3}{5}(10k - 25) - \frac{1}{4}(8 - 16k) = \frac{1}{2}(14k - 10) - 2(k - 9)

What value of kk balances the given equation?

Answer: 6

Answer

The value that satisfies the equation is 6.
The correct value is found by carefully distributing the fractional multipliers and negative signs to eliminate all parentheses, simplifying both sides to get 10k−17=5k+1310k - 17 = 5k + 13, and finally isolating kk by performing balanced operations to find k=6k = 6.

Step-by-Step Solution

1
Distribute the fractional coefficients into the parentheses on both sides of the equation.
(6k−15)−(2−4k)=(7k−5)−2(k−9) (6k - 15) - (2 - 4k) = (7k - 5) - 2(k - 9)
Distribution eliminates the parentheses and clears the fractional multipliers, allowing the terms to be evaluated and combined.
2
Distribute the negative signs for the remaining terms in the equation.
6k−15−2+4k=7k−5−2k+18 6k - 15 - 2 + 4k = 7k - 5 - 2k + 18
Properly applying the negative sign ensures all terms are correctly added or subtracted, especially noting that −14(−16k)=+4k- \frac{1}{4}(-16k) = +4k and −2(−9)=+18-2(-9) = +18.
3
Combine the constant terms and the variable terms independently on each side.
10k−17=5k+13 10k - 17 = 5k + 13
Simplifying each side makes it easier to isolate the variable.
4
Isolate the variable term by moving all terms containing k to one side and constants to the other side.
10k−5k=13+17⇒5k=30⇒k=6 10k - 5k = 13 + 17 \Rightarrow 5k = 30 \Rightarrow k = 6
Solving for k provides the final numerical value that balances the equation.

Key Concept

Solving complex one-variable linear equations by distributing fractional coefficients, managing negative signs, and combining like terms.
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Topics

Solving One-Variable Linear Equations

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