Solve the equation. Express numbers as integers or simplified fractions.

[tex]\frac{6 y-4}{6}=\frac{4 y-8}{9}+\frac{y}{3}[/tex]

To solve the equation 6 6 y − 4 ​ = 9 4 y − 8 ​ + 3 y ​ , we multiply both sides by 18 to eliminate the fractions. This leads to y = − 1 after simplifying. The final answer is y = − 1 .
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Answered by Anonymous | 2025-07-04

Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate fractions: 18 ⋅ 6 6 y − 4 ​ = 18 ⋅ 9 4 y − 8 ​ + 18 ⋅ 3 y ​ .
Simplify the equation: 3 ( 6 y − 4 ) = 2 ( 4 y − 8 ) + 6 y , which expands to 18 y − 12 = 8 y − 16 + 6 y .
Combine like terms and isolate the variable: 18 y − 12 = 14 y − 16 , then 4 y = − 4 .
Solve for y : y = − 1 . The solution set is − 1 ​ .

Explanation

Problem Analysis We are given the equation 6 6 y − 4 ​ = 9 4 y − 8 ​ + 3 y ​ . Our goal is to solve for y .

Eliminating Fractions To eliminate the fractions, we multiply both sides of the equation by the least common multiple (LCM) of the denominators 6, 9, and 3. The LCM of 6, 9, and 3 is 18. Multiplying both sides by 18, we get: 18 ⋅ 6 6 y − 4 ​ = 18 ⋅ 9 4 y − 8 ​ + 18 ⋅ 3 y ​ Simplifying, we have: 3 ( 6 y − 4 ) = 2 ( 4 y − 8 ) + 6 y

Expanding the Equation Next, we expand both sides of the equation: 18 y − 12 = 8 y − 16 + 6 y

Combining Like Terms Combine like terms on the right side of the equation: 18 y − 12 = 14 y − 16

Isolating the Variable Subtract 14 y from both sides: 18 y − 14 y − 12 = 14 y − 14 y − 16 4 y − 12 = − 16

Further Isolation Add 12 to both sides: 4 y − 12 + 12 = − 16 + 12 4 y = − 4

Solving for y Divide both sides by 4: 4 4 y ​ = 4 − 4 ​ y = − 1

Final Answer Therefore, the solution to the equation is y = − 1 . The solution set is { − 1 } .


Examples
Solving linear equations is a fundamental skill in algebra and has numerous real-world applications. For example, suppose you are managing a budget for a school event. You need to determine how many tickets you must sell to cover the costs. By setting up a linear equation that represents the revenue from ticket sales and the expenses for the event, you can solve for the number of tickets needed to break even. This ensures that you can plan effectively and avoid financial losses. Similarly, linear equations are used in physics to calculate motion, in economics to model supply and demand, and in engineering to design structures and circuits.

Answered by GinnyAnswer | 2025-07-04