Problem 48 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ \frac{1}{5}(9 y+2)+\frac{1}{10}(2 y-1) $$

Step-by-Step Solution

Answer

The simplified expression is \(2y + \frac{3}{10}\).

Step 1: Distribute Multiplication

To start simplifying the expression, distribute the fractions \(\frac{1}{5}\) and \(\frac{1}{10}\) to the terms inside each set of parentheses.We distribute \(\frac{1}{5}\) over \(9y+2\):\[\frac{1}{5}(9y+2) = \frac{1}{5} \cdot 9y + \frac{1}{5} \cdot 2 = \frac{9}{5}y + \frac{2}{5}\]Next, distribute \(\frac{1}{10}\) over \(2y-1\):\[\frac{1}{10}(2y-1) = \frac{1}{10} \cdot 2y - \frac{1}{10} \cdot 1 = \frac{2}{10}y - \frac{1}{10} = \frac{1}{5}y - \frac{1}{10}\]

Step 2: Combine Like Terms

Now, combine the expressions obtained from the distribution:\[\frac{9}{5}y + \frac{2}{5} + \frac{1}{5}y - \frac{1}{10}\]Combine the \(y\) terms:\[\left(\frac{9}{5}y + \frac{1}{5}y\right) = \frac{9+1}{5}y = \frac{10}{5}y = 2y\]Then, combine the constant terms:To do this, we convert \(\frac{2}{5}\) to an equivalent fraction with a denominator of 10:\[\frac{2}{5} = \frac{2 \cdot 2}{5 \cdot 2} = \frac{4}{10}\]Now, add \(\frac{4}{10}\) and \(-\frac{1}{10}\):\[\frac{4}{10} - \frac{1}{10} = \frac{3}{10}\]

Step 3: Write the Final Expression

Combine the results from Step 2: the \(y\) terms and the constant terms together to write the simplified expression:\[2y + \frac{3}{10}\]This is the simplified expression with the parentheses removed.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us to simplify expressions by eliminating parentheses. Imagine you have a group of friends, and you want to share candies equally among them. That's pretty much what the distributive property does with numbers—it shares them across terms. In mathematical notation, the property is applied as follows:

• If you have an expression like \( a(b+c) \), you distribute \( a \) by multiplying it with each term inside the parentheses. So, it becomes \( ab + ac \).

In our example, we have fractions that are distributed: \(\frac{1}{5}(9y + 2)\). So, \(\frac{1}{5}\) is multiplied by both \(9y\) and \(2\). That's the trick of distributing fractions, and you do the same with \(\frac{1}{10}(2y - 1)\). We need to distribute these fractions to simplify each part of the expression.
The distributive property makes it easier to handle equations, especially when dealing with more complex algebra involving variables and fractions.

Combining Like Terms

Once you’ve used the distributive property, your next step is to simplify further by combining like terms. Like terms are terms that have the same variable raised to the same power. So, if your mother gave you apples and your father gave you apples, you'll put them all in one basket because they are the same—it is similar in algebra!
For example, when you have expressions such as \( \frac{9}{5}y + \frac{1}{5}y \), both are \( y \) terms. You can add them together because they share the same variable. It's like saying you have 9 apples and 1 more, giving you a total of 10 apples. Hence, \( \frac{9}{5}y + \frac{1}{5}y = \frac{10}{5}y = 2y \).

• Make sure that only alike terms are added, and constants or numbers without variables are handled separately.
• By combining these terms, you create a simpler expression that is easier to interpret and solve.

Mistakes often happen if you try to combine terms that don’t match, so always check the terms carefully!

Fractions in Algebra

Fractions can often be tricky, especially in algebra. But once you get the hang of them, they simply become another tool in your math toolkit! In algebra, we frequently encounter fractions when dealing with the distributive property, just as in our example with the expression:

• We distributed \( \frac{1}{5} \) and handled it by multiplying through the terms inside the parentheses.
• Similarly with \( \frac{1}{10} \), which involves careful multiplication and simplification to ensure the fractions are correctly processed.

Adding or subtracting fractions can require us to find a common denominator. Here, we converted \( \frac{2}{5} \) into a fraction with a denominator of 10, making it \( \frac{4}{10} \). Doing this allows us to easily combine fractions:
\( \frac{4}{10} - \frac{1}{10} = \frac{3}{10} \).
Using common denominators simplifies handling fractions, making it possible to add or subtract them easily, which is essential for solving algebraic expressions efficiently.