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Chapter 5: Problem 31

Express as a single fraction $$ \frac{a_{1}}{s+p_{1}}+\frac{a_{2}}{s+p_{2}} $$

Short Answer

Expert verified
Question: Express the sum of the given fractions as a single fraction: $$\frac{a_{1}}{s+p_{1}} + \frac{a_{2}}{s+p_{2}}$$ Answer: The sum of the given fractions expressed as a single fraction is: $$\frac{a_{1}(s+p_{2})+a_{2}(s+p_{1})}{(s+p_{1})(s+p_{2})}$$

Step by step solution

01

Identify the denominators

Identify the denominators of the given fractions. In this case, they are \((s + p_1)\) and \((s + p_2)\).
02

Find a common denominator

To find a common denominator, we can multiply the two denominators together, as this will always be a multiple of both denominators. So, in this case, the common denominator will be \((s+p_1)(s+p_2)\).
03

Convert fractions to their equivalents

Now that we have a common denominator, we can convert the fractions into their equivalents with the same denominator. We do this by multiplying the numerator and denominator of each fraction by the necessary factor: $$ \frac{a_{1}}{s+p_{1}}\times\frac{s+p_{2}}{s+p_{2}} = \frac{a_{1}(s+p_{2})}{(s+p_{1})(s+p_{2})} \\ \frac{a_{2}}{s+p_{2}}\times\frac{s+p_{1}}{s+p_{1}} = \frac{a_{2}(s+p_{1})}{(s+p_{1})(s+p_{2})} $$
04

Add the equivalent fractions

Now that the fractions have the same denominator, we can add them together by adding their numerators: $$ \frac{a_{1}(s+p_{2})}{(s+p_{1})(s+p_{2})} + \frac{a_{2}(s+p_{1})}{(s+p_{1})(s+p_{2})} = \frac{a_{1}(s+p_{2})+a_{2}(s+p_{1})}{(s+p_{1})(s+p_{2})} $$
05

Final answer

The sum of the given fractions expressed as a single fraction is: $$ \frac{a_{1}(s+p_{2})+a_{2}(s+p_{1})}{(s+p_{1})(s+p_{2})} $$

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Most popular questions from this chapter

Express as a single fraction $$ s+2+\frac{s+3}{(s+1)(s+2)} $$

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