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Chapter 0: Problem 31

Simplify each expression. $$ -12 y+4 y+3 y+2 y $$

Short Answer

Expert verified
-3y

Step by step solution

01

Combine Like Terms

Identify terms that have the same variable, in this case, all terms have the variable y. Combine these terms by adding their coefficients: \(-12y, 4y, 3y, 2y\)
02

Add the Coefficients

Add the coefficients of the like terms together: \(-12 + 4 + 3 + 2\)
03

Simplify the Coefficients

Perform the addition: \(-12 + 4 = -8\) \(-8 + 3 = -5\) \(-5 + 2 = -3\)
04

Write the Final Expression

Combine the result with the variable to get the simplified expression: \(-3y\)

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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 a fundamental skill in algebra that involves reducing an expression to its simplest form. This often means combining like terms. In this process, you aim to make the expression as straightforward as possible by eliminating any unnecessary complexity.
For instance, let's look at the expression \(-12y + 4y + 3y + 2y\). The goal here is to condense this down to a simpler form.

  • First, identify terms that have the same variable. In this case, all terms include the variable y.
  • Next, add or subtract the coefficients of these terms.
  • Finally, combine the result with the variable.
By following these steps, you simplify the expression step by step until it's as simple as possible.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and mathematical operators (like +, -, *, /). They can represent real-world problems and can be manipulated to find solutions for various equations.
In our example, the expression \(-12y + 4y + 3y + 2y\) is an algebraic expression. Here's a breakdown of its components:

  • Numbers: The coefficients -12, 4, 3, and 2.
  • Variables: In this case, the variable is y.
  • Operators: The addition and subtraction signs.
Combining like terms, as we've done here, helps simplify these expressions, making them easier to work with when solving equations or other algebraic problems.
Addition of Coefficients
The addition of coefficients is a crucial part of simplifying algebraic expressions. Coefficients are the numerical parts of the terms that contain variables. When combining like terms, you're adding or subtracting their coefficients.
Taking our example \(-12y + 4y + 3y + 2y\), here's how we add the coefficients:

  • Start with the initial coefficient: -12.
  • Add the next coefficient to it: \-12 + 4 = -8\.
  • Add the following coefficient: \-8 + 3 = -5\.
  • Finally, add the last coefficient: \-5 + 2 = -3\.
This step-by-step addition reduces the complexity and results in the simplified term \(-3y\). Understanding how to properly add and subtract coefficients is key to mastering algebraic simplification.

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Most popular questions from this chapter

The distributive property can be used to mentally perform calculations. $$\begin{array}{l}38 \cdot 17+38 \cdot 3 \\\=38(17+3) \\\=38(20) \\\=760\end{array} $$Use the distributive property to calculate each value mentally. $$ \frac{8}{5}(17)+\frac{8}{5}(13) $$

Evaluate each expression for \(w=4, x=-\frac{3}{4}, y=\frac{1}{2},\) and \(z=1.25 .\) $$ -w+2 x+3 y+z $$

Simplify each expression. $$ 10+30 \div 2 \cdot 3 $$

Simplify each expression. $$ \frac{(-5+\sqrt{4})-2^{2}}{-5-2} $$

Write each fraction as a decimal. For repeating decimals, write the answer by first using bar notation and then rounding to the nearest thousandth. See Example 11. $$ \frac{5}{6} $$
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