Problem 64 - Prealgebra and Introductory Algebra

Question

Is \(-\frac{5}{6}\) a solution of the equation \(\frac{1}{4}+x=-\frac{7}{12} ?\)

Step-by-Step Solution

Answer

Yes, \(-\frac{5}{6}\) is a solution of the equation \(\frac{1}{4}+x=-\frac{7}{12}\)

Step 1: Substitute

Substitute \(-\frac{5}{6}\) for \(x\) in the equation, \(\frac{1}{4}+x=-\frac{7}{12}\). So, the equation turns into \(\frac{1}{4}-\frac{5}{6}\)

Step 2: Simplify

Simplify the left-hand side of the equation: Both \(\frac{1}{4}\) and \(-\frac{5}{6}\) should be converted into similar fractions so that they can be added together. To create like terms, convert \(\frac{1}{4}\) to \(\frac{3}{12}\) and \(-\frac{5}{6}\) to \(-\frac{10}{12}\). Then, \(\frac{3}{12}-\frac{10}{12}=-\frac{7}{12}\)

Step 3: Compare

Compare the simplified left-hand side \(-\frac{7}{12}\) with the right-hand side \(-\frac{7}{12}\). As both are the same, we can conclude that \(-\frac{5}{6}\) is indeed a solution to the equation.

Key Concepts

Understanding Fractions in Equations

When dealing with fractions in algebraic equations, it's essential to know how to manipulate them. Fractions represent parts of a whole and appear frequently in algebra. In this problem, the equation involves two fractions: \(\frac{1}{4}\) and \(-\frac{5}{6}\). To solve or verify the solution of an equation containing fractions, we often must create common denominators.
This allows us to add or subtract them easily. Common denominators transform fractions into like terms, making it simple to carry out operations.
You can find a common denominator by identifying the least common multiple (LCM) of the denominators.

• In our problem, the LCM of 4 and 6 is 12.
• Convert \(\frac{1}{4}\) to \(\frac{3}{12}\) and \(-\frac{5}{6}\) to \(-\frac{10}{12}\).

Once converted, you can perform addition or subtraction straightforwardly.

Solution Verification

Verifying a solution involves checking if the proposed solution satisfies the original equation. After substituting the value of \(x = -\frac{5}{6}\) into the equation, we simplify the expression.
We combined the fractions to calculate the left-hand side: \(\frac{3}{12} - \frac{10}{12} = -\frac{7}{12}\).
Next, compare this with the right-hand side of the equation, which is also \(-\frac{7}{12}\).

• If both sides match, your solution is correct.
• If they don't, the value substituted is not a solution.

This step is crucial in ensuring that the solution fulfills the equation's requirements.

Simplification Process

Simplification is the process of making an equation easier to understand or solve.
It often involves reducing fractions and performing basic arithmetic operations.
In our exercise, we simplified the left side by converting fractions to have a common denominator.

• Reduce fractions: Convert denominators to be the same to perform operations easily.
• Perform operations: Calculate the addition or subtraction of fractions once denominators are alike.

This simplification enabled us to directly compare both sides of the equation. It's a fundamental process that helps clarify complex expressions and provides a clearer path to the solution.