Solve for $w$. $5 w+9 z=2 z+3 w$
Answer (1)
Simplify the equation 5 w + 9 z = 2 z + 3 w by combining like terms.
Isolate w to get 2 w = − 7 z .
Divide by 2 to solve for w , resulting in w = − 2 7 z .
The solution is w = − 2 7 z .
Explanation
Problem Analysis We are given the equation 5 w + 9 z = 2 z + 3 w and four possible solutions for w in terms of z : w = − 2 7 z , w = − 7 2 z , w = − 2 z , and w = − 7 z . Our goal is to solve the equation for w and determine which of the given solutions is correct.
Simplifying the Equation First, let's simplify the given equation by combining like terms. We have:
5 w + 9 z = 2 z + 3 w
Subtract 3 w from both sides:
5 w − 3 w + 9 z = 2 z + 3 w − 3 w
2 w + 9 z = 2 z
Subtract 9 z from both sides:
2 w + 9 z − 9 z = 2 z − 9 z
2 w = − 7 z
Solving for w Now, we solve for w by dividing both sides by 2:
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 see that it matches the first option. Therefore, the correct solution is w = − 2 7 z .
Examples
Understanding how to solve linear equations is crucial in many real-world applications. For instance, consider a scenario where you're trying to determine the relationship between the number of hours worked and the amount earned. If you know that your base pay is related to your total earnings through a linear equation, solving for one variable in terms of another helps you predict your income based on the hours you work, or vice versa. This skill is also applicable in fields like budgeting, where you might need to solve for expenses in terms of income to ensure financial stability.
