Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation. Using the addition property of equality. Be sure to check your proposed solutions. $$x+\frac{7}{8}=\frac{9}{8}$$

Step-by-Step Solution

Answer

The solution of the equation is \(x = \frac{1}{4}\).

Step 1: Understand the Problem

The given equation to solve is \(x+\frac{7}{8}=\frac{9}{8}\). Use the addition property of equality to isolate \(x\).

Step 2: Solve the Equation

To isolate \(x\), subtract \(\frac{7}{8}\) from both sides of the equation. This is the addition property of equality. \(x+\frac{7}{8} - \frac{7}{8} =\frac{9}{8} - \frac{7}{8}\), this simplifies to \(x = \frac{2}{8}\).

Step 3: Simplifying the Solution

Divide the numerator and the denominator of the fraction by the greatest common divisor (GCD). The GCD of 2 and 8 is 2. Thus \(x =\frac{2}{8}/2=\frac{1}{4}\).

Step 4: Check the Solution

Substitute \(x=\frac{1}{4}\) into the original equation \(x+\frac{7}{8}=\frac{9}{8}\) to check if the equality holds. Substituting \(x\) with \(\frac{1}{4}\) gives us: \(\frac{1}{4}+\frac{7}{8}=\frac{2}{8}+\frac{7}{8}=\frac{9}{8}\) so the initial equation holds true and the solution is correct.

Key Concepts

Addition Property of Equality

Solving equations often involves making different operations to both sides, without changing the equation's balance. That's where the **Addition Property of Equality** comes into play. This property tells us that if you have an equation, you can add or subtract the same number on both sides, and the equality will still hold true. In essence, it’s like adjusting the scales in a balanced way! For instance, in the equation \(x + \frac{7}{8} = \frac{9}{8}\), to isolate \(x\), we want to remove \(\frac{7}{8}\) from the left side. By subtracting \(\frac{7}{8}\) from both sides, we maintain balance while simplifying the equation. When subtracting two fractions, it's as if you are moving them around on the number line, maintaining their relationship without changing their overall value.

Fractions in Algebra

Working with fractions can be a bit tricky, but when you break their complexity, it becomes much easier. When you encounter equations with fractions, remember these few tips:

• Make sure the fractions share a common denominator when you perform operations.
• Add or subtract the numerators, as the denominators merely remain in place.
• If necessary, simplify the result by finding the greatest common divisor (GCD).

In our example \(x + \frac{7}{8} = \frac{9}{8}\), the denominators are already the same (they are both 8), allowing for straightforward subtraction. Once we solve the equation for \(x = \frac{2}{8}\), we then simplify it to \(\frac{1}{4}\). This involves dividing the top and bottom by their GCD, which is 2, making the fraction as simple as possible.

Checking Solutions

Once you’ve solved an equation, it's crucial to check your solution. Why? This helps confirm that the solution was calculated correctly and fits the original equation. To do this, simply substitute your solution back into the equation. For example, after finding that \(x = \frac{1}{4}\), replace \(x\) in the original equation \(x + \frac{7}{8} = \frac{9}{8}\). So, substituting gives: \(\frac{1}{4} + \frac{7}{8}\). Convert \(\frac{1}{4}\) to \(\frac{2}{8}\) to match the denominators, making it easier to add: \(\frac{2}{8} + \frac{7}{8} = \frac{9}{8}\). The left-side matches the original right-side, which validates our solution. Checking solutions keeps you from proceeding with incorrect results and ensures confidence in your work.