Select the correct answer.
Which value of $w$ makes this equation true?
$\frac{5 w+4}{3}=\frac{3 w}{2}$
A. -8
B. -2
C. -1
D. 8
Answer (1)
Multiply both sides of the equation by 6 to eliminate the fractions: 2 ( 5 w + 4 ) = 3 ( 3 w ) .
Expand both sides: 10 w + 8 = 9 w .
Subtract 9 w from both sides: w + 8 = 0 .
Subtract 8 from both sides to solve for w : − 8 .
Explanation
Understanding the Problem We are given the equation 3 5 w + 4 = 2 3 w and asked to find the value of w that makes the equation true. We can solve this equation by multiplying both sides by the least common multiple of the denominators, which is 6.
Eliminating Fractions Multiplying both sides of the equation by 6, we get: 6 × 3 5 w + 4 = 6 × 2 3 w 2 ( 5 w + 4 ) = 3 ( 3 w ) Expanding both sides, we have: 10 w + 8 = 9 w
Isolating w Now, we want to isolate w . Subtract 9 w from both sides of the equation: 10 w − 9 w + 8 = 9 w − 9 w w + 8 = 0
Solving for w Finally, subtract 8 from both sides to solve for w :
w + 8 − 8 = 0 − 8 w = − 8
Checking the Answer We can check our answer by substituting w = − 8 back into the original equation: 3 5 ( − 8 ) + 4 = 2 3 ( − 8 ) 3 − 40 + 4 = 2 − 24 3 − 36 = − 12 − 12 = − 12 Since the equation holds true, our solution is correct.
Final Answer The value of w that makes the equation true is w = − 8 . Therefore, the correct answer is A.
Examples
Imagine you're trying to balance a seesaw. The equation 3 5 w + 4 = 2 3 w is like finding the right weight w to put on one side so it perfectly balances with the other side. Solving such equations is crucial in engineering for designing balanced structures, in physics for calculating equilibrium, and even in everyday scenarios like distributing weight in a vehicle to maintain stability. Understanding how to solve these equations helps in making informed decisions to achieve balance and stability in various real-world applications.
