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Chapter 1: Problem 49

Solve the formula for the specified variable. $$E K+L=D-T K \text { for } K$$

Short Answer

Expert verified
\( K = \frac{D - L}{E + T} \).

Step by step solution

01

Rearrange the Equation

Starting with the equation \( EK + L = D - TK \), the first step is to get all terms containing \( K \) on one side. To do this, add \( TK \) to both sides of the equation:\[EK + TK + L = D\]
02

Factor Out the Variable

Now that all terms containing \( K \) are on one side, factor \( K \) from both terms:\[K(E + T) + L = D\]
03

Isolate the Variable

Subtract \( L \) from both sides to isolate the term with \( K \):\[K(E + T) = D - L\]
04

Solve for the Variable

Finally, divide both sides by the factor \( (E + T) \) to solve for \( K \):\[K = \frac{D - L}{E + T}\]

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Key Concepts

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

Rearranging Equations
Rearranging equations is an essential skill that helps us express a given variable in terms of others. In a problem like this, with the equation \( EK + L = D - TK \), the goal is to bring all terms involving the variable \( K \) to one side.
To start, we need to identify which parts of the equation contain our target variable \( K \). Here, it's present in both \( EK \) and \(-TK \). To simplify, we add \( TK \) to both sides, leading to \( EK + TK + L = D \).
This organized approach helps maintain balance in the equation and is a foundation step before proceeding to more complex operations.
Factoring
Factoring is the process of expressing an algebraic expression as a product of its simpler parts. Once we have all terms with the variable \( K \) on one side of the equation \( EK + TK + L = D \), we can factor \( K \) out of its respective terms.
This means rewriting the expression as \( K(E + T) + L = D \).
The factoring isolates \( K \) making it easy to proceed with further manipulations.
This step allows us to see \( K \) more clearly in the equation, setting us up for the isolating step that follows.
Isolating Variables
Isolating variables involves manipulating an equation such that the variable of interest stands alone on one side.
In our rearranged equation, \( K(E + T) + L = D \), we need to get \( K \) by itself.
This requires us to remove the other terms attached to it.
To begin isolating \( K \), subtract \( L \) from both sides of the equation, yielding \( K(E + T) = D - L \).
This step is crucial for clearly defining \( K \) and prepares us to finalize the solution by solving for \( K \).
Precalculus
Precalculus serves as the bridge between arithmetic and the more advanced concepts in calculus. It covers various algebraic concepts, such as solving equations, which involve understanding operations like rearranging, factoring, and isolating variables.
The techniques applied in this problem鈥攔earranging the equation, factoring, and isolating variables鈥攁re precalculus skills that provide the foundation needed for tackling calculus problems.
These steps ensure a strong grasp of algebra before moving on to more complex mathematical challenges that calculus introduces, such as understanding limits and derivatives.

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Most popular questions from this chapter

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O'Carroll's formula is used to handicap weight lifters. If a lifter who weighs \(b\) kilograms lifts \(w\) kilograms of welght, then the handicapped weight \(W\) is given by $$W=\frac{w}{\sqrt{b-35}}$$ Suppose two lifters weighing 75 kilograms and 120 kilograms lift weights of 180 kilograms and 250 kilograms, respectively. Use O'Carroll's formula to determine the superior weight lifter.

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The basal energy requirement for an individual indicates the minimum number of calories necessary to maintain essential life-sustaining processes such as circulation, regulation of body temperature, and respiration. Given a person's sex, weight \(w\) (in kilograms), height \(h\) (in centimeters), and age \(y\) (in years), we can estimate the basal energy requirement in calories using the following formulas, where \(C_{f}\) and \(C_{m}\) are the calories necessary for females and males, respectively: $$\begin{array}{l}C_{f}=66.5+13.8 w+5 h-6.8 y \\\C_{m}=655+9.6 w+1.9 h-4.7 y\end{array}$$ (a) Determine the basal energy requirements first for a 25 -year-old female weighing 59 kilograms who is 163 centimeters tall and then for a 55 -year-old male weighing 75 kilograms who is 178 centimeters tall. (b) Discuss why, in both formulas, the coefficient for \(y\) is negative but the other coefficients are positive.
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