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Chapter 6: Problem 46

Solve: \(x+\frac{2}{5}=\frac{9}{10}\).

Short Answer

Expert verified
\( x = \frac{1}{2} \)

Step by step solution

01

Subtract \(\frac{2}{5}\) from both sides

To isolate the variable \(x\), subtract \(\frac{2}{5}\) from both sides of the equation.\[x + \frac{2}{5} - \frac{2}{5} = \frac{9}{10} - \frac{2}{5}\]
02

Simplify the right side

Find a common denominator for the fractions on the right side. The common denominator for \(\frac{9}{10}\) and \(\frac{2}{5}\) is 10.\[ \frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10} \]Now substitute: \[ \frac{9}{10} - \frac{4}{10} = \frac{5}{10} \]
03

Simplify the fraction

Simplify \(\frac{5}{10}\).\[ \frac{5}{10} = \frac{1}{2} \]
04

Write the final solution

Now that the right side is simplified, the equation is: \[ x = \frac{1}{2} \]

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Key Concepts

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

Fractions
Working with fractions often intimidates students, but understanding them is crucial for solving many algebraic equations. A fraction represents a part of a whole and consists of a numerator (top number) and a denominator (bottom number). For example, in the fraction \(\frac{2}{5}\), 2 is the numerator and 5 is the denominator. When fractions appear in equations, we need to perform operations like addition, subtraction, multiplication, or division while considering both the numerators and denominators. For our exercise, knowing how to manipulate fractions correctly allowed us to isolate the variable.
Common Denominator
When dealing with multiple fractions, finding a common denominator is essential for performing addition or subtraction. A common denominator is a shared multiple of the denominators of two or more fractions. In our exercise, we had to subtract \(\frac{2}{5}\) from \(\frac{9}{10}\). The common denominator for 5 and 10 is 10 since 10 is the smallest number that both 5 and 10 can divide into without leaving a remainder.
The process to find a common denominator helps to simplify the fractions so they can be easily added or subtracted. For instance, for the fraction \(\frac{2}{5}\), we converted it to \(\frac{4}{10}\) to match the denominator of \(\frac{9}{10}\).
Variable Isolation
Variable isolation is a fundamental concept in solving equations. It involves rearranging an equation to solve for an unknown variable. For instance, in the equation \(\frac{2}{5} + x = \frac{9}{10}\), the goal is to isolate x. To achieve this, we perform inverse operations to both sides of the equation to keep it balanced.
In our exercise, we subtracted \(\frac{2}{5}\) from both sides to isolate x. The equation transformed step-by-step until we found the solution \(\frac{1}{2}\).
The process of isolating the variable ensures that we systematically narrow down to the exact value of the unknown, making it clear and straightforward.

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Most popular questions from this chapter

Solve. California Sales Tax. The sales tax rate in California is \(7.25 \%\). How much sales tax is charged on a purchase of 4 contour foam travel pillows at \(\$ 39.95\) each? What is the total price?

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Convert to standard notation. $$ 42.7 \text { million } $$
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