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Chapter 1: Problem 6

Solve for the indicated variable in each formula. \(a t=x-b t\) solve for \(t\)

Short Answer

Expert verified
\(t = \frac{x}{a + b}\)

Step by step solution

01

Isolate terms involving t

Start by moving all terms containing the variable \(t\) to one side of the equation. The original equation is \(a t = x - b t\). Add \(b t\) to both sides to get: \[a t + b t = x\]
02

Factor out t

Now factor out \(t\) from the left-hand side of the equation. The equation \(a t + b t = x\) becomes: \[t(a + b) = x\]
03

Solve for t

To solve for \(t\), divide both sides of the equation by \((a + b)\), assuming \(a + b eq 0\): \[t = \frac{x}{a + b}\]

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Key Concepts

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

Solving Equations
Equations are mathematical statements that assert the equality of two expressions. Solving equations involves finding the values of the variables that make the equation true. In the given exercise, our goal is to determine the value of the variable \( t \) in terms of other given quantities.When solving equations, you'll typically follow these steps:
  • Identify the variable you need to solve for, known as the 'unknown'.
  • Use algebraic operations to rearrange the equation, so the variable appears on one side alone.
  • Simplify the equation as much as possible to find a solution for the unknown.
In this exercise, you highlight each instance of \( t \) and work to combine them on one side.
Formula Manipulation
Formula manipulation involves rearranging formulas to express one variable in terms of others. This skill is crucial for solving physics problems, converting units, or isolating specific variables.In our exercise, you start with the formula \( a t = x - b t \). The manipulation involves:
  • Adding \( b t \) to both sides to gather all terms involving \( t \) together.
  • Simplifying the equation to combine similar terms, resulting in \( a t + b t = x \).
  • Recognizing that both terms on the left share the variable \( t \), which allows for factoring.
This step-by-step manipulation enables the rearrangement of the original equation into a more usable form, paving the way to isolate \( t \).
Variable Isolation
Variable isolation is a method used to express a variable explicitly in terms of others. This approach is essential in solving equations and understanding relationships between different variables in algebra.In our problem, after formula manipulation, we reach the equation \( t(a + b) = x \). Here鈥檚 how to isolate \( t \):
  • Recognize that \( t \) is multiplied by the quantity \( (a + b) \).
  • To isolate \( t \), divide both sides of the equation by \( (a + b) \), assuming that \( a + b eq 0 \).
  • You'll find \( t = \frac{x}{a + b} \), which expresses \( t \) solely in terms of \( a \), \( b \), and \( x \).
By isolating the variable, the problem is solved, providing a clear expression with \( t \) fully separated from other terms.

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Most popular questions from this chapter

Find all values of \(x\) that make the equation true. $$|x+5|=-2$$

Simplify the algebraic fraction. $$\frac{(x+1)^{3}(3 x-5)-(x+1)^{2}(8 x+3)}{(x-4)(x+1)^{3}}$$

Use scientific notation and the laws of exponents to perform the indicated operations. Give the result in scientific notation rounded to two significant figures. $$\frac{\left(1 \times 10^{-3}\right)\left(3.28 \times 10^{6}\right)}{4 \times 10^{7}}$$

Determine whether each statement is true for all real numbers \(x\). If the statement is false, then indicate one counterexample, i.e. a value of \(x\) for which the statement is false. $$-x \leq 0$$

Write an inequality to represent the given interval and state whether the interval is closed, open or half-open. Also state whether the interval is bounded or unbounded. $$[-5,3]$$
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