Problem 33 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{6-4}{9-2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{2}{7}\).

Step 1: Simplify the Numerator

Begin by simplifying the numerator of the fraction. The numerator is the expression \(6 - 4\). Subtract \(4\) from \(6\) to get \(2\). So, the simplified numerator is \(2\).

Step 2: Simplify the Denominator

Next, simplify the denominator of the fraction. The denominator is the expression \(9 - 2\). Subtract \(2\) from \(9\) to get \(7\). So, the simplified denominator is \(7\).

Step 3: Write the Simplified Expression

After simplifying both the numerator and the denominator, rewrite the fraction with these simplified values. The simplified expression becomes \(\frac{2}{7}\). This fraction is already in its simplest form.

Key Concepts

Understanding the Numerator

The numerator is an essential part of a fraction. It represents the top number in a fraction. When you have a fraction like \( \frac{a}{b} \), the numerator is \( a \). In this exercise, the numerator was \(6 - 4\). To simplify it, it's important to perform the operation indicated. Subtraction is about taking away the second number from the first. So, when you subtract \(4\) from \(6\), you are left with \(2\). Therefore, in this context, the simplified numerator is \(2\). Understanding how to simplify the numerator means knowing how to perform basic operations accurately, an essential skill for dealing with fractions.

Understanding the Denominator

Similarly to the numerator, the denominator is crucial in a fraction. It appears below the numerator at the bottom of the fraction. In the fraction \( \frac{a}{b} \), \( b \) is the denominator. In our exercise, the denominator started as \(9 - 2\). To simplify this, you also perform a subtraction. You subtract \(2\) from \(9\), resulting in \(7\). Thus, the simplified denominator becomes \(7\). It's important to remember that the denominator indicates the number of equal parts the whole is divided into. Simplifying the denominator correctly is vital because it maintains the integrity of the fraction's meaning and ensures calculations are precise.

Subtracting Integers

Subtracting integers can sometimes be tricky, but having a clear understanding makes it easier. An integer is a whole number that can be positive, negative, or zero. When you subtract an integer, you are finding the difference between two numbers. In the expression \(6 - 4\), you are effectively removing \(4\) parts from \(6\), resulting in \(2\). Another example is \(9 - 2\), where you remove \(2\) from \(9\), leading to \(7\). Here are some tips to keep in mind:

• Always look for the larger number and subtract the smaller number from it.
• Visualize number lines, which can help if you're unsure about results.
• Remember, if the number you subtract is larger, be prepared to end up with a negative result.

Understanding subtraction helps not only with simplifying numerators and denominators but also lays the groundwork for more complex arithmetic functions.