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Chapter 7: Problem 22

To solve the following equation for \(r,\) what is the first step? $$ r p-r q=p+q $$

Short Answer

Expert verified
The first step is to combine like terms on one side of the equation.

Step by step solution

01

Identify Common Terms

Observe that the term involving the variable, r, appears on both sides of the equation. To simplify, collect all the r terms on one side.
02

Combine Like Terms

Move the r term on the left side to the right. The equation can be rewritten as: \( r p - r q = p + q \).
03

Factor Out the Common Variable

Factor out r from the left side of the equation. This gives us: \( r(p - q) = p + q \).
04

Solve for r

Divide both sides by (p - q) to isolate r. Thus, \( r = \frac{p + q}{p - q} \).

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Key Concepts

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

Combining Like Terms
In algebra, combining like terms is an essential skill. It simplifies equations and makes them easier to solve. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(3x + 4x - 2x\), all terms are like terms because they all include the variable \(x\).

To combine like terms, simply add or subtract the coefficients. In our exercise, we initially have the equation \(r p - r q = p + q\).

The term \(r\) appears on both sides, so the first step is to collect or combine these like terms. We convert the equation to \(r(p - q) = p + q\). This unifies the terms involving \(r\), making further steps much more manageable.
Factoring
Factoring is an important strategy to further simplify equations. It's the process of expressing something as a product of its factors. In our example, after combining like terms, we ended up with \(r p - r q = p + q\). To isolate \(r\), we need to factor it out.

By factoring out the common variable \(r\), we get \((r)(p - q) = p + q\). Factoring makes this equation easier to solve because it isolates the variable \(r\) on one side.

Remember that recognizing common factors and restarting the equation in a factored form helps isolate variables effectively.
Isolating Variables
Isolating variables is a key step in solving equations. The goal is to get the variable you are solving for on one side of the equation while everything else is on the opposite side.

In our equation, after factoring, we had \(r(p - q) = p + q\). To solve for \(r\), divide both sides by \(p - q\). This gives us \(r = \frac{p + q}{p - q}\).

Isolating the variable makes it clear what the variable equals. This process is fundamental in algebra and is frequently used in solving linear equations, rational expressions, and more.
Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Solving equations involving rational expressions often requires clear simplification and isolation of variables.

In our solved equation, \(r = \frac{p + q}{p - q}\) can be seen as a rational expression. The goal was to solve for \(r\), which involved cleaning up terms and factoring.

Rational expressions often appear in algebra, requiring a solid understanding of operations like addition, subtraction, multiplication, division, and factoring. Practicing these skills makes solving equations involving rational expressions more approachable.

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Most popular questions from this chapter

Multiply or divide as indicated. $$ \frac{(6 x+5)(x-3)}{(x-1)(x-3)} \div \frac{(2 x+7)(x+9)}{(x-1)(x+9)} $$

Multiply or divide as indicated. $$ \left(\frac{6 k^{2}-13 k-5}{k^{2}+7 k} \div \frac{2 k-5}{k^{3}+6 k^{2}-7 k}\right) \cdot \frac{k^{2}-5 k+6}{3 k^{2}-8 k-3} $$

Solve each problem. See Examples \(1-7\) The amount of light (measured in foot-candles) produce. by a light source varies inversely as the square of the distance from the source. If the illumination produced \(1 \mathrm{m}\) from a light source is 768 foot-candles, find the illumination produced \(6 \mathrm{m}\) from the same source.

Solve each problem mentally. Use proportions in Exercises 23 and 24. In a mathematics class, 3 of every 4 students are girls. If there are 28 students in the class, how many are girls? How many are boys?

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{7 x+8}{3 x+2}-\frac{x+4}{3 x+2} $$
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