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Chapter 1: Problem 81

Solve for the specified variable. $$ A=\frac{1}{3}\left(s_{1}+s_{2}+s_{3}\right) \quad \text { for } s_{3} $$

Short Answer

Expert verified
\( s_3 = 3A - s_1 - s_2 \)

Step by step solution

01

Understand the formula

The formula provided is for the average of three scores, where \( A \) is the average, and \( s_1, s_2, \) and \( s_3 \) are the scores. Our task is to solve for \( s_3 \).
02

Isolate the average term

Since the average is given by \( A = \frac{1}{3}(s_1 + s_2 + s_3) \), we need to eliminate the fraction. Multiply both sides of the equation by 3 to get:\[ 3A = s_1 + s_2 + s_3 \]
03

Solve for \( s_3 \)

Now that we have \( 3A = s_1 + s_2 + s_3 \), we can solve for \( s_3 \) by subtracting \( s_1 \) and \( s_2 \) from both sides: \[ s_3 = 3A - s_1 - s_2 \]
04

Simplify the expression (if necessary)

The expression \( s_3 = 3A - s_1 - s_2 \) is already simplified and is the solution for \( s_3 \) in terms of \( A, s_1, \) and \( s_2 \).

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Key Concepts

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

Understanding Variables in Equations
In algebra, a variable is a symbol that represents a value that can change or hold different values. It is typically represented by letters such as \(x\), \(y\), or any other symbol like \(s_1\), \(s_2\), and \(s_3\) in our example. Understanding variables is crucial when working to solve equations because they allow us to express mathematical relationships in general terms.
In the given equation \(A = \frac{1}{3}(s_1 + s_2 + s_3)\), the variables \(s_1, s_2, s_3\), and \(A\) represent numbers or measurements. We need to manipulate these variables to solve for \(s_3\). The context often provides meaning to these variables, in this case, scores, and the average of those scores.
The Art of Algebraic Manipulation
Algebraic manipulation involves rearranging equations to isolate a specific variable. This process is essential for solving equations or making computations simpler. In our example, we start with the equation \(A = \frac{1}{3}(s_1 + s_2 + s_3)\). The goal is to isolate \(s_3\).
  • First, eliminate the fraction by multiplying both sides by 3: \(3A = s_1 + s_2 + s_3\). This step aligns with the principle of maintaining equality by performing the same operation on both sides of the equation.
  • Next, to solve for \(s_3\), subtract \(s_1\) and \(s_2\) from both sides: \(s_3 = 3A - s_1 - s_2\).
This step is crucial as it gives us an expression for \(s_3\) only in terms of \(A, s_1,\) and \(s_2\). The result is a clean solution highlighting the importance of organizing and simplifying equations.
Applying the Average Formula
The average formula is used to find the mean of a set of numbers. In simple terms, it's the sum of the numbers divided by the count of the numbers. When we have three scores, the average is expressed as \(A = \frac{s_1 + s_2 + s_3}{3}\).
This formula is applied in many fields, like statistics, to find a data set's central value. In the context of the problem, the formula helps determine \(s_3\) when the values for \(A, s_1,\) and \(s_2\) are known.
  • The first step in using the average formula is to recognize that dividing the sum by 3 gives the average.
  • Multiplying the average by 3 recovers the original sum \(s_1 + s_2 + s_3\).
  • Subsequently, isolating \(s_3\) involves further straightforward steps, as shown earlier.
Understanding this process expands the ability to tackle a variety of problems involving averages, whether in mathematics, sciences, or finance.

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