Problem 20 - Elementary and Intermediate Algebra

Question

Add and simplify the result, if possible. \(\frac{y+2}{10 z}+\frac{y+4}{10 z}\)

Step-by-Step Solution

Answer

\(\frac{y+3}{5z}\)

Step 1: Identify Common Denominator

Both fractions have the same denominator, which is \(10z\). So, we can directly add the numerators.

Step 2: Add the Numerators

Since the denominators are the same, add the numerators: \((y + 2) + (y + 4) = 2y + 6\).

Step 3: Combine the Fractions

Place the sum of the numerators over the common denominator: \(\frac{2y + 6}{10z}\).

Step 4: Simplify the Expression

Check if the expression can be simplified. Factor the numerator: \(2(y + 3)\), and the denominator is \(10z = 2 imes 5 imes z\). Cancel the common factor of 2: \(\frac{y+3}{5z}\).

Key Concepts

Adding Fractions

When we encounter fractions, especially in algebra, adding them can be a tad intimidating at first. However, the key to mastering this is finding a common denominator. The denominator is the number at the bottom of the fraction and it needs to be the same for both fractions when you add them.
For example: If you have \( \frac{y+2}{10z} \) and \( \frac{y+4}{10z} \), you immediately notice they have the same denominator. This makes it simpler, as you can directly add their numerators. Numerators are the top parts of the fractions.

• Find a common denominator - it's already there!
• Add the numerators together: \((y + 2) + (y + 4) = 2y + 6\).

With a common denominator established, you can just focus on the numerators and add them together!

Simplifying Expressions

After adding fractions, often you will want to simplify the resulting expression. Simplifying means making it as simple as possible to work with or understand.
When you have combined the numerators and placed them over the common denominator, you get the new fraction: \( \frac{2y + 6}{10z} \). In algebra, you should always check whether this new fraction can be reduced to a simpler form.

• Look at the numerator: \(2y + 6\). Can you pull out common factors?
• Yes! The common factor is \(2\), so you factor it out to get \(2(y + 3)\).
• Look at the denominator: \(10z = 2 \times 5 \times z\). Notice the common factor \(2\) in the numerator and the denominator.

By canceling common factors in the numerator and denominator, we simplify fractions into a more manageable form. The expression ultimately becomes \( \frac{y+3}{5z}\).
So remember, simplifying not only cleans up the expression but also makes further calculations easier!

Factoring Polynomials

Factoring is a crucial skill in algebra. It involves breaking down expressions into their simplest parts or 'factors'. When simplifying expressions with fractions, factoring helps to identify common elements. This makes the process of simplifying much easier.
For example, \(2y + 6\) can look complex at first glance. But notice both terms, \(2y\) and \(6\), share a common factor: \(2\). You can write the expression as \(2(y + 3)\). This is what we call factoring out the greatest common factor (GCF).

• Identify any common factor in the terms.
• Extract that factor out of the entire expression: \(2(y + 3)\).

On the denominator side with \(10z = 2 \times 5 \times z\), we used the factor \(2\) to simplify the fraction.
Using factoring in this way ensures that your expression is in its simplest possible form, ready for whatever algebraic challenges lie ahead!