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Chapter 11: Problem 51

Solve. $$t^{-1}=\frac{1}{2}[7.6]$$

Short Answer

Expert verified
The solution is \( t = 0.263 \).

Step by step solution

01

- Simplify the Right Side

First, simplify the expression on the right side of the equation. Since \(\frac{1}{2}[7.6]\) equals \(\frac{7.6}{2}\), compute the division: \[ \frac{7.6}{2} = 3.8 \]
02

- Rewrite the Equation

Rewrite the original equation with the simplified right side: \[ t^{-1} = 3.8 \]
03

- Solve for t

Take the reciprocal of both sides to solve for \( t \): \[ t = \frac{1}{3.8} \]
04

- Perform the Division

Finally, compute the value of \( t \) by performing the division: \[ t = 0.263 \]

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Key Concepts

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

reciprocals
In mathematics, a reciprocal is simply the inverse of a number. For a non-zero number, the reciprocal is what you multiply that number by to get 1. Mathematically, the reciprocal of a number x is represented as: \[ x^{-1} = \frac{1}{x} \] For example, the reciprocal of 5 is \( \frac{1}{5} \). Reciprocals are especially useful when solving equations, as they help us isolate variables. In our exercise, we used the reciprocal to solve for the variable \( t \).
algebraic equations
An algebraic equation is a mathematical statement of equality between two expressions. It typically contains variables, numbers, and arithmetic operations. The goal of solving an algebraic equation is to find the value of the variable that makes the equation true. For example, in the exercise \( t^{-1} = 3.8 \), we aim to find a value of \( t \) where this equation holds true. By performing operations on both sides of the equation, we can isolate \( t \) and determine its value.
simplification
Simplification in equations involves reducing expressions to their most basic form. This can make them easier to handle and solve. In our exercise, we first simplified the expression \( \frac{1}{2}[7.6] \) to \( \frac{7.6}{2} \), and then further simplified it to 3.8. Simplifying helps in getting a clearer, more manageable equation that we can solve step-by-step. It's a crucial step in solving algebraic equations efficiently.
division
Division is one of the four fundamental arithmetic operations. It is essentially the process of determining how many times one number is contained within another. In the context of our exercise, division played a key role. We divided 7.6 by 2 to simplify the right side of the equation. Finally, we performed a division operation again to find the value of \( t \) as \( \frac{1}{3.8} \), which resulted in approximately 0.263. Understanding and performing division correctly is vital in solving equations.

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Most popular questions from this chapter

If \(f(x)=\left(x-\frac{1}{3}\right)\left(x^{2}+6\right)\) and \(g(x)=\) \(\left(x-\frac{1}{3}\right)\left(x^{2}-\frac{2}{3}\right),\) find all \(a\) for which \((f+g)(a)=0\).

Let \(f(x)=x^{2} .\) Find \(x\) such that \(f(x)=11\).

Solve. $$ 7 x^{2}=21 $$

Solve. $$ 9 x^{2}-16=0 $$

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ f(x)=x^{2}+6 x+7 $$
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