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Chapter 6: Problem 20

Solve each equation, and check your solutions. \(\frac{4}{y}+\frac{1}{y}=2\)

Short Answer

Expert verified
The solution is \( y = \frac{5}{2} \).

Step by step solution

01

Combine Like Terms

Combine the fractions on the left side of the equation: evaluate \(\frac{4}{y}+\frac{1}{y}\) to get \(\frac{4+1}{y}\).This simplifies to \(\frac{5}{y} = 2\).
02

Eliminate the Denominator

Multiply both sides of the equation by y to eliminate the denominator: \(\frac{5}{y} \times y = 2 \times y\). This simplifies to 5 = 2y.
03

Solve for y

Isolate y by dividing both sides by 2: \( y = \frac{5}{2} \).
04

Check the Solution

Substitute y with \(\frac{5}{2}\) in the original equation to check the result: \( \frac{4}{\frac{5}{2}} + \frac{1}{\frac{5}{2}} = 2\).Simplifying each term gives: \( \frac{4 \times 2}{5} + \frac{1 \times 2}{5} = 2\).This resolves to: \( \frac{8}{5} + \frac{2}{5} = 2\).Finally, \(\frac{10}{5} = 2\) confirms the solution is correct.

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Key Concepts

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

Combining Like Terms
Combining like terms is an essential skill when solving rational equations. It means grouping terms that have the same variables raised to the same power. In the initial step of solving the equation \[ \frac{4}{y} + \frac{1}{y} = 2 \], we see two fractions with the same denominator, y.
To combine these terms, you add the numerators together: \[ \frac{4}{y} + \frac{1}{y} = \frac{4 + 1}{y} = \frac{5}{y} \]. This simplifies the equation to just one fraction on the left side.
Learning to recognize like terms makes solving equations more straightforward. Always look for identical denominators or matching variable parts to combine them effectively.
Eliminating Denominators
Eliminating denominators helps simplify equations, making them easier to solve. In our example, \[ \frac{5}{y} = 2 \], the denominator y complicates the equation.
The simplest way to eliminate it is to multiply every term by y. So, you multiply both sides: \[ \frac{5}{y} \times y = 2 \times y \], which simplifies to 5 = 2y.
Now, you have an equation without any fractions. Always remember, multiplying both sides by the same value won't change the equality, but it will often simplify the equation considerably.
Checking Solutions
After solving an equation, it's vital to check your solutions to ensure they're correct. Substituting your answer back into the original equation helps confirm this.
In our example equation, the solution was y = \[ \frac{5}{2} \]. Substitute y back into the original to check: \[ \frac{4}{\frac{5}{2}} + \frac{1}{\frac{5}{2}} = 2 \].
You simplify each fraction: \[ \frac{4 \times 2}{5} + \frac{1 \times 2}{5} = 2 \], which simplifies further to \[ \frac{8}{5} + \frac{2}{5} = 2 \].
Combining these gives \[ \frac{10}{5} = 2 \], which is a true statement. This confirms our solution is correct. Always substitute back to avoid errors and ensure your solution works for the original equation.

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Most popular questions from this chapter

Divide. Write each answer in lowest terms. $$ \frac{(x-3)^{2}}{6 x} \div \frac{x-3}{x^{2}} $$

Only one of these choices is equal to \(\frac{\frac{1}{3}+\frac{1}{12}}{\frac{1}{2}+\frac{1}{4}} .\) Which one is it? Answer this question without showing any work, and explain your reasoning. A. \(\frac{5}{9}\) B. \(-\frac{5}{9}\) C. \(-\frac{9}{5}\) D. \(-\frac{1}{12}\)

Write each rational expression in lowest terms. $$ \frac{q^{2}-4 q}{4 q-q^{2}} $$

Rewrite each rational expression with the indicated denominator. $$ \frac{6}{k^{2}-4 k}=\frac{?}{k(k-4)(k+1)} $$

Match each division problem in Column I with the correct quotient in Column II. (a) \(\frac{5 x^{3}}{10 x^{4}} \div \frac{10 x^{7}}{4 x}\) (b) \(\frac{10 x^{4}}{5 x^{3}} \div \frac{10 x^{7}}{4 x}\) (c) \(\frac{5 x^{3}}{10 x^{4}} \div \frac{4 x}{10 x^{7}}\) (d) \(\frac{10 x^{4}}{5 x^{3}} \div \frac{4 x}{10 x^{7}}\) A. \(\frac{5 x^{5}}{4}\) B. \(5 x^{7}\) C. \(\frac{4}{5 x^{5}}\) D. \( \frac{1}{5 x^{7}}\)
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