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Chapter 3: Problem 91

Solve. \(A=\frac{T+Q}{2},\) for \(Q\)

Short Answer

Expert verified
Q = 2A - T

Step by step solution

01

Rewrite the Equation

Rewrite the given equation to make it easier to solve: \[ A = \frac{T + Q}{2} \]
02

Eliminate the Denominator

Multiply both sides of the equation by 2 to eliminate the denominator: \[ 2A = T + Q \]
03

Isolate Q

Subtract T from both sides of the equation to isolate Q: \[ 2A - T = Q \]
04

Solution

Express the isolated term Q clearly: \[ Q = 2A - T \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Manipulation
Algebraic manipulation is a key concept when solving equations. It's the process of rearranging and simplifying equations to make finding the solution easier. Here are some techniques:
  • Combining like terms: Group similar terms together to simplify the equation.
  • Using inverse operations: Apply operations that undo each other, like adding and subtracting or multiplying and dividing.
  • Balancing equations: Perform the same operation on both sides of the equation to keep it balanced.
In our exercise, algebraic manipulation was used to eliminate the fraction. We multiplied both sides by 2 to remove the denominator. This step is crucial because it simplifies the equation, making it easier to work with in subsequent steps.
Isolating Variables
Isolating variables is the process of getting the unknown variable alone on one side of the equation. This allows you to solve for that variable. Here鈥檚 how it's done step-by-step:
  • Move terms involving the variable: Use addition or subtraction to move terms to the other side.
  • Clear coefficients: Use division or multiplication to get the variable by itself.
In our example, the goal was to solve for Q. After simplifying the equation by eliminating the denominator, we focused on getting Q alone by subtracting T from both sides. Successfully isolating Q provided us with the solution: \( Q = 2A - T \)
Equation Solving Steps
Solving an equation involves a series of logical steps. Each step brings you closer to finding the unknown variable. Let's break down the steps used in our exercise:
  • Rewrite the equation: This ensures we understand it fully and set it up for easier manipulation.
  • Eliminate the fraction: By multiplying every term by 2, we simplify and move closer to isolating the variable.
  • Isolate the variable: Through subtraction, we moved all non-Q terms to the other side of the equation.
  • Express the solution: The final step shows our isolated variable clearly and concisely.
Following these steps systematically and patiently is essential. They ensure no mistakes and help you understand each part of the process.

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