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Difficulty: HardSolving One-Variable Linear Equations

A linear equation is structured as follows:

12(4y−6)−23(9−6y)=2y+15 \frac{1}{2}(4y - 6) - \frac{2}{3}(9 - 6y) = 2y + 15

Based on this relationship, what is the correct value of yy?

  1. A
    -6
  2. B
    -3
  3. C
    3
  4. 6Answer

Answer

6
The correct value is found by properly distributing the fractional coefficients, carefully tracking negative signs, and preserving equality to isolate the variable. The operations simplify the initial equation to 6y - 9 = 2y + 15, which reduces to 4y = 24, resulting in 6.

Step-by-Step Solution

1
Distribute 12 \frac{1}{2} into the first set of parentheses.
2y−3 2y - 3
Applying the distributive property eliminates the first group.
2
Distribute −23 -\frac{2}{3} into the second set of parentheses.
−6+4y -6 + 4y
Multiplying the negative fraction across the terms. Notice that multiplying two negative values (−2/3-2/3 and −6y-6y) yields a positive result (+4y+4y).
3
Combine the distributed terms on the left side of the equation.
(2y+4y)+(−3−6)=6y−9 (2y + 4y) + (-3 - 6) = 6y - 9
Combining like terms simplifies the left side of the equation before attempting to isolate the variable.
4
Set the simplified left side equal to the right side and solve for y.
6y−9=2y+15⇒4y−9=15⇒4y=24⇒y=6 6y - 9 = 2y + 15 \Rightarrow 4y - 9 = 15 \Rightarrow 4y = 24 \Rightarrow y = 6
Using inverse operations groups the variable terms on one side and constant terms on the other to reveal the final value.

Key Concept

Solving complex linear equations involving fractional distribution and sign management.
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Topics

Solving One-Variable Linear Equations

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