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Chapter 6: Problem 19

Solve each formula for the specified variable. \(R=\frac{g s}{g+s} ; g\)

Short Answer

Expert verified
g = \frac{R s}{s - R}

Step by step solution

01

Introduce the Formula

The given formula is \[R = \frac{g s}{g + s}\]and we need to solve for the variable \(g\).
02

Remove the Fraction

To eliminate the fraction, multiply both sides by \(g + s\). This gives:\[R(g + s) = g s\]
03

Distribute R

Distribute \(R\) on the left-hand side:\[R g + R s = g s\]
04

Collect Like Terms

Isolate the terms involving \(g\) on one side of the equation. Move \(R g\) to the right side:\[R s = g s - R g\]Then factor out the \(g\) on the right side:\[R s = g(s - R)\]
05

Isolate g

Divide both sides by \(s - R\) to solve for \(g\):\[g = \frac{R s}{s - R}\]

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Key Concepts

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

Formula Manipulation
Understanding formula manipulation is key in solving equations like the one given in the exercise. The idea is to rearrange the formula to isolate a particular variable.

Let's break it down:
  • First, identify the variable to solve for. In our exercise, it's the variable g.
  • Next, use algebraic operations to move other terms around, keeping track of operations like addition, subtraction, multiplication, and division.

The process requires careful attention to each step to ensure accuracy and maintain the balance of the equation.
Isolating Variables
Isolating variables involves moving all terms containing the desired variable to one side of the equation, and everything else to the other side. Here's a deeper dive:

  • Start by multiplying out any fractions. In our example, we multiplied both sides by \(g + s\) to clear the fraction.
  • Next, distribute any multiplicands, as seen when \(R(g + s) = gs\) became \(Rg + Rs = gs\).
  • Then, collect like terms. For our exercise, we moved \(Rg\)\to the right side to get \(Rs = gs - Rg\).

Finally, factor out the desired variable and isolate it by dividing by the remaining terms. In the exercise, we factored and isolated g to get \[g = \frac{Rs}{s - R}\].
Algebraic Fractions
Algebraic fractions feature a numerator and a denominator that are algebraic expressions. Solving equations with algebraic fractions can be tricky, but here are some key steps:

  • Clear the fraction by multiplying both sides by the denominator. In our exercise, multiplying by \(g + s\) helped to get rid of the fraction.
  • Once the fraction is cleared, the equation simplifies into standard algebraic terms.
  • Distribute and combine like terms as needed to make solving easier, just like when we combined \(g\) terms in the example.

Getting comfortable with these steps helps in solving more complex algebraic fractions in various contexts.

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Most popular questions from this chapter

Find an equation of variation in which: \(y\) varies directly as \(x\) and inversely as \(w\) and the square of \(z,\) and \(y=4.5\) when \(x=15, w=5,\) and \(z=2\).

A device used in golf to estimate the distance \(d\) to a hole measures the size \(s\) that the 7 -ft pin appears to be in a viewfinder. The viewfinder uses the principle, diagrammed here, that \(s\) gets bigger when \(d\) gets smaller. If \(s=0.56\) in. when \(d=50\) yd, find an equation of variation that expresses \(d\) as a function of \(s .\) What is \(d\) when \(s=0.40\) in.?

The harmonic mean of two numbers \(a\) and \(b\) is a number \(M\) such that the reciprocal of \(M\) is the average of the reciprocals of \(a\) and \(b .\) Find a formula for the harmonic mean.

To check Example \(4,\) Kara graphs $$ y_{1}=\frac{7 x^{2}+21 x}{14 x} \text { and } y_{2}=\frac{x+3}{2} $$ since the graphs of \(y_{1}\) and \(y_{2}\) appear to be identical, Kara believes that the domains of the functions described by \(y_{1}\) and \(y_{2}\) are the same, \(\mathbb{R} .\) How could you convince Kara otherwise?

Use a graphing calculator to check Example 3 by setting \(y_{1}=\left(2 x^{2}-7 x-15\right) /(x-5)\) and \(\left.y_{2}=2 x+3 . \text { Then use either ( table }\right)\) (after selecting the ZOOM ZINTEGER option) or ( table \()\) (with TblMin \(=0\) and \(\Delta \mathrm{Tbl}=1\) ) to show that \(y_{1} \neq y_{2}\) for \(x=5\).
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