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Chapter 5: Problem 36

Simplify each expression by combining like terms. See Examples 6 through 10. $$ 5 y+7 y-6 y $$

Short Answer

Expert verified
The expression simplifies to \(6y\).

Step by step solution

01

Identify Like Terms

In the given expression \(5y + 7y - 6y\), we first identify like terms. Here, all the terms contain the variable \(y\). Therefore, \(5y\), \(7y\), and \(-6y\) are all like terms.
02

Combine Like Terms

To simplify, we combine these like terms by adding and subtracting their coefficients. We have: \(5y + 7y - 6y = (5 + 7 - 6)y\).
03

Calculate the Coefficients

Now, perform the arithmetic with the coefficients: \(5 + 7 = 12\)\(12 - 6 = 6\).Thus, the simplified expression for the coefficients is \(6\).
04

Write the Simplified Expression

Bring together the simplified coefficients and the shared variable:The simplified expression is \(6y\).

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Key Concepts

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

Simplifying Algebraic Expressions
Simplifying algebraic expressions is a key skill in algebra. It involves rewriting an expression in its simplest form. Imagine an expression like a messy room that needs tidying. By simplifying it, we make it more organized and easier to understand.

To simplify an expression, you first need to gather and combine all the terms that look alike. This process is called combining like terms. Once the like terms are grouped, perform arithmetic operations on their coefficients, and finally rewrite the expression.

Simplifying helps in solving equations faster. It also provides a clearer understanding of the relationships between different parts of the expression.
Identifying Like Terms
Identifying like terms is the first step in the process of simplifying an expression. Like terms in an algebraic expression have the same variable raised to the same power, but they can have different coefficients. It's like finding matching socks in a laundry pile.

For example, in the expression \(5y + 7y - 6y\), each term has the variable \(y\) with an exponent of 1. Thus, they are like terms. Different variables or different powers of the same variable would mean they're not like terms.

Remember these key points about like terms:
  • They must have identical variables.
  • The variables must have the same exponents.
  • Only like terms can be combined to simplify expressions.
Arithmetic with Coefficients
Once you've identified and grouped like terms, it's time to do arithmetic with the coefficients. Coefficients are numerical factors that multiply the variables in an algebraic expression.

Using the expression from earlier, \(5y + 7y - 6y\), the coefficients are 5, 7, and -6. To simplify, you add or subtract these coefficients while keeping the common variable. The operation would be: \((5 + 7 - 6)y\).

By performing the arithmetic, you get \(12 - 6\), which equals 6. Hence, the simplified expression becomes \(6y\).
  • Addition is commutative, so the order of like terms doesn鈥檛 affect the result.
  • Only the coefficients are combined; the variable and its power remain unchanged.
Understanding this concept is crucial because it underpins all algebraic simplification and solving techniques.

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Most popular questions from this chapter

Subtract \(\left(4 x^{2}-2 x+2\right)\) from the sum of \(\left(x^{2}+7 x+1\right)\) and \((7 x+5)\)

Subtract. $$ (5 x+8)-\left(-2 x^{2}-6 x+8\right) $$

Add or subtract as indicated. $$ (3 x+5)+(2 x-14) $$

Add or subtract as indicated. $$ (7 y+7)-(y-6) $$

Simplify each expression by combining any like terms. $$ y-10+y $$
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