Problem 65 - Algebra 2

Question

Simplify each difference. $$ \frac{3 y+1}{4 y+4}-\frac{2 y+7}{2 y+2} $$

Step-by-Step Solution

Answer

The simplified form of the difference is \( \frac{-y - 13}{4(y + 1)} \)

Step 1 - Simplify the Fractions

First, simplify each fraction separately by factoring out the common factors in the numerator and the denominator, giving \( \frac{3y + 1}{4(y + 1)} - \frac{2y + 7}{2(y + 1)} \)

Step 2 - Finding Common Denominator

Next, to subtract fractions, they should have a common denominator. So find the common denominator which is \(4(y + 1)\) and rewrite the fractions with that common denominator, giving \( \frac{3y + 1}{4(y + 1)} - \frac{2(2y + 7)}{4(y + 1)} \) or \( \frac{3y + 1}{4(y + 1)} - \frac{4y + 14}{4(y + 1)} \)

Step 3 - Subtracting Numerators

The denominator is now the same, meaning we can subtract the numerators. So, the result becomes, \(\frac{(3y + 1)-(4y + 14)}{4(y + 1)} \)

Step 4 - Simplify the Result

Lastly, simplify the fraction you get from the subtraction, which gives \( \frac{-y - 13}{4(y + 1)} \)

Key Concepts

Simplifying Fractions

Do you know that simplifying fractions can make working with them much easier? Let's break it down. Simplifying a fraction means reducing it to its simplest form by dividing the numerator and the denominator by their greatest common factor (GCF).
For example, in the original exercise, we have two fractions linked by subtraction:

• \(\frac{3y + 1}{4y + 4}\)
• \(\frac{2y + 7}{2y + 2}\)

Let's start with the first fraction, \(\frac{3y + 1}{4y + 4}\). We can simplify it by factoring the denominator: \(4y + 4\) becomes \(4(y + 1)\). Now see how there's no common factor to factor out with the numerator \(3y + 1\), so it stays the same.
Next, the second fraction \(\frac{2y + 7}{2y + 2}\) can be simplified by factoring the denominator too: \(2y + 2\) simplifies to \(2(y + 1)\). Again, there’s no simpler form for the numerator \(2y + 7\). Understanding this process of simplification sets the stage to handle these fractions confidently in your math journey.

Common Denominator

To subtract fractions, they need to have the same bottom number, or denominator. When the denominators are different, finding a common denominator is necessary. This will help us perform the subtraction easily.
Consider our simplified fractions:

• \(\frac{3y + 1}{4(y + 1)}\)
• \(\frac{2y + 7}{2(y + 1)}\)

To find a common denominator, we look for the least common multiple (LCM) of the denominators \(4(y + 1)\) and \(2(y + 1)\). It turns out, the LCM is \(4(y + 1)\) because it accounts for both denominators adequately.
Now, adjust each fraction to have this common denominator. The first fraction already has \(4(y + 1)\). The second one needs to be multiplied by 2 (its missing portion)value in both the top and bottom parts to equal \(\frac{4y + 14}{4(y + 1)}\). By setting equivalent denominators, subtraction becomes straightforward.

Subtracting Fractions

Once both fractions have a common denominator, subtracting them becomes simple. You subtract only the numerators and keep the common denominator unchanged.
Given the expressions from our previous sections with a common denominator of \(4(y + 1)\):

• \(\frac{3y + 1}{4(y + 1)}\)
• \(\frac{4y + 14}{4(y + 1)}\)

Subtract the numerators: \((3y + 1) - (4y + 14)\). Simplify the result:

• Combine like terms to get: \(-y - 13\)

The final, simplified difference is \(\frac{-y - 13}{4(y + 1)}\).
By using a common denominator, you tackle the subtraction of fractions easily, avoiding common pitfalls and gaining confidence in handling these operations in math.