Solve for $w$.
$5 w+9 z=2 z+3 w$
Answer (1)
Simplify the equation 5 w + 9 z = 2 z + 3 w .
Combine like terms to get 2 w = − 7 z .
Divide both sides by 2 to isolate w .
The solution is w = − 2 7 z , so the final answer is − 2 7 z .
Explanation
Problem Analysis We are given the equation 5 w + 9 z = 2 z + 3 w and the possible solutions for w are w = − 2 7 z , w = − 7 2 z , w = − 2 z , w = − 7 z . Our goal is to solve for w in terms of z and identify the correct solution from the given options.
Simplifying the Equation First, let's simplify the given equation by combining like terms: 5 w + 9 z = 2 z + 3 w
Isolating w Subtract 3 w from both sides of the equation: 5 w − 3 w + 9 z = 2 z + 3 w − 3 w
2 w + 9 z = 2 z
Continue Isolating w Next, subtract 9 z from both sides of the equation: 2 w + 9 z − 9 z = 2 z − 9 z 2 w = − 7 z
Solving for w Now, divide both sides by 2 to solve for w :
2 2 w = 2 − 7 z w = − 2 7 z
Identifying the Correct Solution Comparing our solution w = − 2 7 z with the given options, we find that it matches the first option.
Final Answer Therefore, the correct solution is w = − 2 7 z .
Examples
Understanding how to solve for one variable in terms of another is crucial in many real-world applications. For example, in physics, you might need to express the velocity of an object in terms of time and acceleration. Similarly, in economics, you might want to express the price of a product in terms of supply and demand. These types of algebraic manipulations allow us to model and understand complex relationships between different quantities.
