Problem 22 - Prealgebra

Question

Solve each equation. $$a+\frac{1}{4}=-\frac{3}{4}$$

Step-by-Step Solution

Answer

a = -1

Step 1: Isolate the Variable

To solve the equation \(a + \frac{1}{4} = -\frac{3}{4}\), first isolate \(a\). Do this by subtracting \(\frac{1}{4}\) from both sides. The equation becomes: \[a = -\frac{3}{4} - \frac{1}{4}\]

Step 2: Simplify the Right Side

Now, simplify the right side of the equation. You're going to perform the subtraction: \[-\frac{3}{4} - \frac{1}{4} = -\frac{4}{4}\]Since \(-\frac{4}{4}\) simplifies to \(-1\), the equation becomes:\[a = -1\]

Key Concepts

Isolating Variables

When working with prealgebra equations, the concept of isolating variables is crucial. Isolating the variable means you need to get the variable by itself on one side of the equation. This allows you to determine its value without any other numbers or variables in the way. In the given equation: \[a + \frac{1}{4} = -\frac{3}{4}\], we need to isolate \(a\) by getting rid of the fraction \(\frac{1}{4}\) added to it. To achieve this, perform the inverse operation, which is subtraction, on both sides of the equation. By subtracting \(\frac{1}{4}\) from both sides, the equation transforms to: \[a = -\frac{3}{4} - \frac{1}{4}\]. This step ensures \(a\) is by itself on the left side, making it easier to solve for its value. Always remember that whatever operation you perform on one side should be echoed on the other to maintain the balance of the equation. This is a fundamental rule in solving equations.

Equation Simplification

Once you have isolated the variable, the next step is simplifying the equation. Simplification makes equations more manageable and usually involves performing basic arithmetic operations. In our problem, after isolating the variable, we have: \[a = -\frac{3}{4} - \frac{1}{4}\]. To simplify this, you combine the fractions on the right side. Since they have the same denominator, you can directly subtract the numerators: \(-3\) and \(-1\). Thus, the subtraction goes as follows: \[-3 - 1 = -4\]. This results in the fraction \(-\frac{4}{4}\), which simplifies to \(-1\). So the equation becomes: \[a = -1\]. Simplifying equations not only provides the solution but also helps verify that all steps are followed correctly, ensuring the problem is accurately solved.

Fraction Subtraction

Subtracting fractions can sometimes seem confusing, but it follows a straightforward rule. When subtracting fractions with the same denominator, subtract only the numerators and keep the denominator the same. In this exercise, the fractions \(\frac{1}{4}\) and \(-\frac{3}{4}\) share the denominator of \(4\). To subtract these fractions:

• Keep the denominator \(4\) unchanged.
• Subtract the numerators: \(-3 - 1\), which equals \(-4\).

Hence, the result is \(-\frac{4}{4}\). This fraction further simplifies to \(-1\), as the numerator and denominator divide out to give \(-1\). Whenever subtracting fractions, always ensure they have a common denominator first. Once they do, subtract the numerators, and, if possible, simplify the resulting fraction to its simplest form. This practice is essential for working through more complex prealgebra problems effectively.