Subtracting Algebraic Expressions Word Problems

Concept

In mathematics, just like we subtract many numbers as we can and find the difference, we subtract two or many algebraic expressions too. However, for the subtraction of algebraic expressions, we combine all the like terms and then subtract them.

Like terms are the terms that have the same power for the same variables. In like terms, one can only change the numerical coefficient. Terms that have different variables or the same variables raised to different powers are known as, unlike terms.

Word problems that include subtraction of algebraic expressions can be solved with ease if we know how to subtract them. We need to write out an algebraic expression that we can evaluate to find our answer. To write our algebraic expression, we carefully read the problem to figure out what our important numbers are and what kind of operation we are dealing with.

Rules

There are two methods to do the subtraction of algebraic expressions:
Horizontal method of Subtraction of Algebraic Expressions
Column method for Subtraction of Algebraic Expressions

Horizontal method of Subtraction of Algebraic Expressions
1. Write all the expressions in a horizontal line by putting them into brackets and put a subtraction sign in between.
2. Change the subtraction to addition by using the additive inverse.
2. Group all the like terms together from all the expressions and rewrite the expression so formed.
3. Add numerical coefficients of all the like terms followed by the common variable.
4. Rewrite the simplified expression, and make sure all the terms in the final answer should be unlike terms.

Column method for Subtraction of Algebraic Expressions
1. Change the subtraction to addition by using the additive inverse.
2. Write all the expressions one below the other . Make sure to like terms in one column. If there a term whose like term is not there in the second expression, then either write below it or leave that column blank.
3. Add the numerical coefficient of each column (like terms) and write below it in the same column followed by the common variable.

Example

The number of pepperoni pizza ordered in one evening is represented by 3x + 1. The number of cheese pizza ordered is represented by 6x – 2. How many more cheese pizzas were ordered than pepperoni, if = 4?

Solution

1. Write the expression showing cheese pizzas minus pepperoni pizzas.
(6x – 2) – (3x + 1)

2. Group by like terms and use additive inverses to solve.
6x – 2 + (-3x – 1) The additive inverse of 3x + 1 is (-3x – 1)
= (6x – 3x) + (-2 – 1)
= 3x – 3

When x = 4,
3x – 3
3(4) – 3
12 – 3 = 9

9 more cheese pizzas were ordered than pepperoni.

Practice Subtracting Algebraic Expressions Word Problems

Practice Problem 1

Carol is (2x – 1) years old. Her brother Gary is (x + 1) years younger than Carol. Their aunt Helen is 6 times as old as Gary. Write and simplify an expression that represents Helen’s age. Find the number that represents Helen’s age if x = 7.

Practice Problem 2

Write a linear expression in the simplest form to represent the side of the triangle if the perimeter is represented by (13x – 1) centimeters.

Find the side of the triangle if x = 3.

Practice Problem 3

Ruth’s number of stickers is represented by $\fn_phv {\color{Teal} \frac{1}{2}}$ (8x + 2). She gave (x + 1) to her friend Mark. Write and simplify the expression that represents the remaining number of Ruth’s stickers. Find the remaining number of Ruth’s stickers if x = 6.

Linear Expression – an algebraic expression in which the variable is raised to the first power, and variables are not multiplied or divided.

Term – either a single number or variable, or numbers and variables multiplied together. Terms are separated by + or − signs.

Like terms – Terms that have the same power for the same variables. In like terms, one can only change the numerical coefficient.

Unlike terms – Terms that have different variables or the same variables raised to different powers.

Distributive Property – to multiply a sum or difference by a number, multiply each term inside the parenthesis by the number outside the parenthesis.

Constant – a term without a variable.

Variable – In algebra, a symbol (usually a letter) standing in for an unknown numerical value in an equation or expression.

Coefficient – is an integer that is multiplied with the variable of a single term or the terms of a polynomial.

Pre-requisite Skills
Evaluate Variable Expressions-addition and subtraction
Evaluate Expressions (Multiplication and Division)
Solve Algebraic Equations – Addition and Subtraction
Solve Algebraic Equations – Multiplication and Division
Evaluate Algebraic Expressions
Write Algebraic Expressions
The Distributive Property
Simplifying Complex Algebraic Expressions
Add Linear Expressions