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Chapter 4: Problem 10

Solve each equation. Check your proposed solution. $$ \frac{1}{2}+a=-\frac{3}{8} $$

Short Answer

Expert verified
The solution is \( a = -\frac{7}{8} \).

Step by step solution

01

Isolate the variable

To solve for the variable \( a \), we need to isolate it on one side of the equation. Start by subtracting \( \frac{1}{2} \) from both sides of the equation to move it to the other side: \[ a = -\frac{3}{8} - \frac{1}{2} \]
02

Simplify the equation

To subtract \( \frac{1}{2} \) from \( -\frac{3}{8} \), we need a common denominator. The least common denominator between 8 and 2 is 8. Rewrite \( \frac{1}{2} \) as \( \frac{4}{8} \). Substitute to get: \[ a = -\frac{3}{8} - \frac{4}{8} \] Now, combine the fractions: \[ a = -\frac{3 + 4}{8} = -\frac{7}{8} \]
03

Check the solution

Substitute \( a = -\frac{7}{8} \) back into the original equation to ensure it is correct: \[ \frac{1}{2} + (-\frac{7}{8}) = -\frac{3}{8} \] Convert \( \frac{1}{2} \) to \( \frac{4}{8} \) and add: \[ \frac{4}{8} - \frac{7}{8} = -\frac{3}{8} \] Both sides of the equation match, so the solution is verified.

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Key Concepts

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

Common Denominator
When working with fractions, a common denominator is crucial for performing operations like addition and subtraction. A common denominator is the same denominator used for two or more fractions, making them easier to compare or combine.
  • To find a common denominator, identify the least common multiple (LCM) of the denominators you have.
  • In our exercise, the fractions involved were \(\frac{1}{2}\) and \(-\frac{3}{8}\). The denominators are 2 and 8.
  • The least common multiple of 2 and 8 is 8. Therefore, we will convert \(\frac{1}{2}\) into \(\frac{4}{8}\), since it is equivalent and simplifies the operation.
Finding a common denominator is the essential first step for simplifying and solving equations involving fractions. By doing this, we can perform arithmetic operations on fractions smoothly.
Fraction Operations
Fraction operations involve arithmetic actions like addition, subtraction, multiplication, or division of fractions.
Let鈥檚 look at subtraction, which is relevant to the exercise:
  • Once we have a common denominator, we can directly subtract the numerators.
  • In the problem, we subtracted \(-\frac{4}{8}\) from \(-\frac{3}{8}\) which results in \(-\frac{7}{8}\). The denominator remains the same while the numerators are subtracted.
  • If you need to add or subtract fractions without a common denominator, convert them to have one, as demonstrated with \(\frac{1}{2}\) from the exercise.
Remembering these steps ensures that you handle fractions appropriately during any algebraic manipulation.
Verifying Solutions
After finding a potential solution, verification is an important final step. This ensures the solution is correct and satisfies the original equation.
  • To verify, substitute the solution back into the original equation. Convert any fractions as needed.
  • In our exercise, substitute \(a = -\frac{7}{8}\) back, giving us \(\frac{1}{2} + (-\frac{7}{8})\).
  • Convert \(\frac{1}{2}\) to \(\frac{4}{8}\) and perform the operation: \(\frac{4}{8} - \frac{7}{8} = -\frac{3}{8}\).
  • The left side equates to the right side, confirming that \(a = -\frac{7}{8}\) is indeed the correct solution.
Verification is particularly important in math as it helps verify any possible arithmetic errors and ensures the process is correctly applied.

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Most popular questions from this chapter

Solve. In your own words, explain how to round a mixed number to the nearest whole number.

Simplify the following. $$\frac{1-\frac{x}{4}}{2+\frac{3}{8}}$$

Perform the indicated operations. $$ -4 \frac{2}{5} \cdot 2 \frac{3}{10} $$

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{65}{12}$$

The Gauge Act of 1846 set the standard gauge for U.S. railroads at \(56 \frac{1}{2}\) inches. (See figure.) If the standard gauge in Spain is \(65 \frac{9}{10}\) inches, how much wider is Spain's standard gauge than the U.S. standard gauge? (Source: San Diego Railroad Museum)
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