Chapter 7.3, Problem 21HP

To determine

Explain how to add linear expressions without using numbers in your explanation.

Explanation of Solution

Given:

Explain how to add linear expressions without using numbers in your explanation.

Concept Used:

Like terms: When we look at algebraic terms to find like terms, first we ignore the coefficients and only look if terms have the same variables with same exponents. Those terms which qualify this condition are called like terms.

All the given four terms $4a,19a,-2a,-3a$ are like terms, because each of them have the same single variable 'a'

Here are the basic steps to follow to simplify an algebraic expression:

Remove parentheses by multiplying factors.

Use exponent rules to remove parentheses in terms with exponents.

Combine like terms by adding coefficients.

Combine the constants.

Addition of Algebraic expression In addition of algebraic expressions while adding algebraic expressions we collect the like terms and add them. The sum of several like terms is the like term whose coefficient is the sum of the coefficients of these like terms.

Example:

1. Add: 6a + 8b, 2b - 4a

Solution: 

(6a + 8b) + (2b - 4a) = 6a + 8b + 2b - 4a

Arrange the like terms together, then add. Thus, the required addition

= 6a - 4a + 8b + 2b =2a + 10b

Subtraction of Algebraic Expression:

To subtract two or more monomials that are like terms, subtract the coefficients; keep the variables and exponents on the variables the same.

In subtraction of like terms when all the terms are negative, subtract their coefficients, also the variables and power of the like terms remains the same.

Rule for subtracting the linear expression: $(a+b)-(c-d)=a+d-c+d$ (change the sign of the 2^nd expression $(c-d)$ which is subtracted from the 1^st expression $(a+b)$ ).

Example:

  $\begin{array}{l}(12a+7b)-(6a-9b)\\ 12a+7b-6a+9b[\text{change}\text{the}\text{signs}\text{of}\text{the}\text{exp}(6a-9b)]\\ 12a-6a+7b+9b[\text{add}\text{the}\text{like}\text{terms}]\\ 6a+16b\\ (12a+7b)-(6a-9b)=6a+16b\end{array}$