Solve for $z$.
$\frac{x-y+z}{4}=w$
Answer (2)
Multiply both sides of the equation by 4: x − y + z = 4 w .
Add y and subtract x from both sides to isolate z : z = 4 w − x + y .
The solution for z is: z = 4 w − x + y .
Explanation
Understanding the Problem We are given the equation 4 x − y + z = w and our goal is to isolate z on one side of the equation. This involves using algebraic manipulations to move all other terms to the other side.
Multiplying by 4 First, we multiply both sides of the equation by 4 to eliminate the fraction: 4 × 4 x − y + z = 4 × w This simplifies to: x − y + z = 4 w
Isolating z Next, we want to isolate z . To do this, we add y to both sides and subtract x from both sides of the equation: x − y + z − x + y = 4 w − x + y This simplifies to: z = 4 w − x + y
Examples
In physics, this type of equation can be used to solve for a variable in a system where multiple factors contribute to a final result. For example, if w represents the average velocity of an object, x represents the initial position, y represents the displacement, and z represents the final position, then the equation 4 x − y + z = w can be rearranged to find the final position z given the other variables. This is a fundamental concept in kinematics, where understanding the relationship between position, displacement, and average velocity is crucial for analyzing motion.
To solve for z in the equation 4 x − y + z = w , first multiply both sides by 4 to eliminate the fraction. Then, rearrange the equation to isolate z, yielding z = 4 w − x + y . This method demonstrates the systematic approach needed to manipulate algebraic equations to find specific variables.
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