Problem 24 - Prealgebra

Question

Solve each equation. $$x-\frac{7}{8}=\frac{3}{8}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{5}{4}\).

Step 1: Identify the Equation

The given equation is \(x - \frac{7}{8} = \frac{3}{8}\). This is a simple linear equation.

Step 2: Isolate the Variable x

To isolate \(x\), add \(\frac{7}{8}\) to both sides of the equation. This will help in moving the constant term to the right side.\[x - \frac{7}{8} + \frac{7}{8} = \frac{3}{8} + \frac{7}{8}\]

Step 3: Simplify the Equation

On the left side, the \(-\frac{7}{8}\) and \(+\frac{7}{8}\) cancel each other, leaving just \(x\). On the right side, combine the fractions:\[x = \frac{3}{8} + \frac{7}{8}\]

Step 4: Add the Fractions

Add the fractions by adding the numerators directly, as the denominators are the same.\[x = \frac{3 + 7}{8}\]\[x = \frac{10}{8}\]

Step 5: Reduce the Fraction

Simplify \(\frac{10}{8}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2.\[x = \frac{10 \div 2}{8 \div 2}\]\[x = \frac{5}{4}\]

Step 6: Final Answer

The solution to the equation is \(x = \frac{5}{4}\).

Key Concepts

Isolating Variables

When solving linear equations, isolating the variable means getting the variable, like \(x\), all by itself on one side of the equation. This is an essential step as it reveals the value of the variable you're solving for.
For the equation \(x - \frac{7}{8} = \frac{3}{8}\), isolating \(x\) involves moving the fraction \(-\frac{7}{8}\) from the left side to the right. This is done by performing the opposite operation to eliminate the fraction next to \(x\). Since the fraction is subtracted from \(x\), add \(\frac{7}{8}\) to both sides. This keeps the equation balanced. Here's how it looks:
\[x - \frac{7}{8} + \frac{7}{8} = \frac{3}{8} + \frac{7}{8}\]
After this step, the negative and positive \(\frac{7}{8}\) cancel each other out, effectively isolating \(x\). Now, you can proceed to solve the remaining fraction on the other side.

Adding Fractions

Adding fractions is a fundamental skill in algebra, especially when fractions share the same denominator. To add fractions, simply add the numerators and place the result over the common denominator.
In\(x = \frac{3}{8} + \frac{7}{8}\), both fractions have a denominator of \(8\). Therefore, combine the numerators \(3\) and \(7\) directly.
\[x = \frac{3 + 7}{8}\]
So it becomes \(x = \frac{10}{8}\).

• Keep the denominator the same - it tells you how many equal parts make a whole.
• Focus only on the numerators, since they count the parts you have.

It's important to simplify your answer if possible, which involves our next concept: simplifying fractions.

Simplifying Fractions

Simplifying fractions helps make them easier to read and understand. It involves reducing a fraction to its simplest form, where numerator and denominator share no common factors other than 1.
The fraction \(\frac{10}{8}\) can be simplified by dividing the numerator and denominator by their greatest common divisor (GCD). Here, the GCD of \(10\) and \(8\) is \(2\).

• Divide \(10\) by \(2\): \(10 \div 2 = 5\).
• Divide \(8\) by \(2\): \(8 \div 2 = 4\).

This simplifies \(\frac{10}{8}\) to \(\frac{5}{4}\).
Remember, a simplified fraction is simply a different expression of the same value, often more concise and easier to work with in calculations.