Problem 61 - Prealgebra

Question

Apply the distributive property, then simplify. $$-3\left(\frac{2}{3} y+\frac{5}{6}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-2y - \frac{5}{2}\).

Step 1: Apply the Distributive Property

To apply the distributive property, multiply the term outside the parentheses by each term inside the parentheses separately.Using the distributive property: \[-3 \cdot \left(\frac{2}{3}y + \frac{5}{6}\right) = -3 \cdot \frac{2}{3}y + (-3) \cdot \frac{5}{6}\]

Step 2: Simplify Each Term

Simplify each term by performing the multiplication:- For \(-3 \cdot \frac{2}{3}y\):\[-3 \cdot \frac{2}{3}y = -\frac{6}{3}y = -2y\]- For \((-3) \cdot \frac{5}{6}\):\[-3 \cdot \frac{5}{6} = -\frac{15}{6} = -\frac{5}{2} \]

Step 3: Combine the Results

Write the combined expression from the results of the distributed multiplication.Combine the terms:\[-2y - \frac{5}{2}\]

Key Concepts

Simplification of Algebraic Expressions

Simplifying algebraic expressions is like tidying up a room. You want everything to be in its simplest form so it's easy to work with. When you start with an expression, it might have multiple terms and operations, but by simplifying, you reduce it to its most basic components. In the exercise provided, we first applied the distributive property to eliminate the parentheses. Each term in the parentheses was multiplied by -3, the number outside the parentheses. After distributing, the expression looked complicated with fractions involved. However, simplifying doesn't end there; it also involves tidying up the fractions and constants, so they are easy to read and work with.
- In general, simplifying expressions means dealing with negatives and fractions through multiplication and division to make them simpler.- The goal is often to reach a form that is easy for further calculations or analysis.
Reaching the -2y and -\(\frac{5}{2}\) parts in our solution comes from simplifying each distributed term individually. This allows us to see the true values each part of the expression represents.

Multiplication of Fractions

Multiplying fractions is a fundamental skill in simplifying algebraic expressions. When multiplying a whole number by a fraction, convert the whole number into a fraction by giving it a denominator of 1. This makes multiplication straightforward since you can multiply numerators together and denominators together. For the term -\(3 \cdot \frac{2}{3}y\) in the exercise, the multiplication resulted in -6 on the numerator (since -3 times 2 is 6) and 3 on the denominator. Simplifying this gives us -2, affecting the "y" in the expression to give us -2y.

• Always try simplifying fractions at the end of a multiplication, as fractions can often be reduced.
• Cross-cancel when you can to make calculations easier before completing the multiplication.

For the other term, \((-3) \cdot \frac{5}{6}\), the process is similar. The multiplication gives -15 over 6, which simplifies further to -\(\frac{5}{2}\). Multiplying fractions often results in this need to simplify further, turning complex numbers into manageable portions.

Combining Like Terms

After distributing and simplifying, combining like terms is the next step to make the expression even cleaner. It involves gathering all terms of the same "type" together. "Like terms" are terms that have the same variable raised to the same power, as well as constants (numbers without variables). In our expression, the terms to combine are the ones left after distribution: -2y and -\(\frac{5}{2}\) don't combine since they are not "like" ones. Combining is often used when separate terms share variables or powers.

• Constant terms combine with each other because they lack variables.
• Similar variable terms combine by adding their coefficients, which are the numbers attached to the variables.

In this scenario, no further combination was needed because -2y and -\(\frac{5}{2}\) are distinguished by their formats: one involving "y" and the other being just a number. Recognizing when terms are unlike and can't combine aids in achieving a correct, finished solution.