Solve for $y: 2(3 y+5)=3\left(5 y+\frac{1}{3}\right)$. The solution is $y=$
Answer (1)
Expand both sides of the equation: 2 ( 3 y + 5 ) = 6 y + 10 and 3 ( 5 y + 3 1 ) = 15 y + 1 .
Set the expanded expressions equal: 6 y + 10 = 15 y + 1 .
Isolate y by subtracting 6 y and 1 from both sides: 9 = 9 y .
Solve for y by dividing both sides by 9 : y = 1 .
Explanation
Problem Setup We are given the equation 2 ( 3 y + 5 ) = 3 ( 5 y + 3 1 ) and we need to solve for y .
Expanding Both Sides First, we expand both sides of the equation: 2 ( 3 y + 5 ) = 6 y + 10 3 ( 5 y + 3 1 ) = 15 y + 1
Setting Equal Now we set the expanded expressions equal to each other: 6 y + 10 = 15 y + 1
Isolating y Next, we want to isolate y on one side of the equation. We can subtract 6 y from both sides: 10 = 9 y + 1
Further Isolating y Now, subtract 1 from both sides: 9 = 9 y
Solving for y Finally, divide both sides by 9 to solve for y :
y = 9 9 = 1
Final Answer Thus, the solution is y = 1 .
Examples
In real-world scenarios, solving linear equations like this can help determine the break-even point in business. For example, if y represents the number of units you need to sell, the equation could represent the point where the cost equals the revenue. Solving for y tells you how many units you need to sell to break even. Understanding how to manipulate and solve these equations is crucial for making informed decisions in various fields, from finance to engineering.
