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Chapter 4: Problem 69

Simplify the expressions. $$\frac{1}{2} y+\frac{3}{2} y$$

Short Answer

Expert verified
2y

Step by step solution

01

Combine Like Terms

Rewrite the expression by identifying like terms. The given expression is \(\frac{1}{2} y + \frac{3}{2} y\). Both terms have the common variable \(y\), so they can be combined.
02

Add the Coefficients

Add the coefficients (the numerical parts) of the like terms. \( \frac{1}{2} + \frac{3}{2} = \frac{4}{2} \).
03

Simplify the Fraction

Simplify the fraction by dividing the numerator by the denominator: \( \frac{4}{2} = 2 \)
04

Write the Final Expression

Multiply the simplified coefficient by the common variable. The final expression is \( 2y \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simplifying expressions
Simplifying expressions is like cleaning up a messy room. We want to make the expression as simple as possible. The first step is to look for like terms, which are terms that have the same variables raised to the same powers. In our example, both \( \frac{1}{2} y \) and \( \frac{3}{2} y \) are like terms because they both contain \( y \).
Once we've identified the like terms, we combine them by adding or subtracting their coefficients.
  • Group the like terms together.
  • Add or subtract the coefficients.
This makes it easier to see the simplified form of the expression. By focusing on these steps, you鈥檒l discover that simplifying expressions is easy and straightforward.
coefficients
In mathematics, coefficients are the numbers in front of the variables. Think of them as the amount you have of something. For example, in \( \frac{1}{2} y \), the coefficient is \( \frac{1}{2} \). It tells us how many \( y \)'s we have.
  • A coefficient can be positive or negative.
  • It can be a whole number, a fraction, or a decimal.
When combining like terms, we only add or subtract the coefficients and keep the variable part the same.
In our example, we add the coefficients \( \frac{1}{2} \) and \( \frac{3}{2} \) to get \( \frac{4}{2} \), which simplifies to 2. This means we have 2 \( y \)'s.
fractions
Fractions represent parts of a whole. In our problem, we dealt with \( \frac{1}{2} \) and \( \frac{3}{2} \). To add or subtract fractions, you need a common denominator. Luckily, our fractions had the same denominator (2), so we could add them directly.
  • Add the numerators together while keeping the denominator the same.
  • Simplify if possible by dividing the numerator and the denominator by the same number.
\( \frac{1}{2} + \frac{3}{2} = \frac{4}{2} \). Upon simplifying \( \frac{4}{2} \), we get 2.
This makes it clear that operations involving fractions can be quite simple when they share a common denominator.

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