In mathematics, an algebraic expression is a combination of variables and constants, as well as algebraic operations (addition, subtraction, multiplication, and division). Terms are the building blocks of an algebraic expression.
There are some algebraic expressions in which the like terms in the expression have the same variables raised to the same exponents. The like terms are constant. To find the terms and the like terms in an expression, we first write it as a sum of its terms first.
When an algebraic expression has neither like terms, nor parentheses, it is said to be in its simplest form. To add or subtract coefficients that are associated to variables, we use the distributive property.
In mathematics, an expression is a sentence that contains at least two numbers and one math operation.
We can use several expressions to understand the manner in which we can identify parts of an expression:
Expression 1: Consider the following expression: 7a + 5a – 21 + 7
We have to identify the terms as well as the like terms in the given expression.
Answer:
Rewrite the expression as a sum of the terms.
7a + 5a – 21 + 7
= 7a + 5a + (-21) + 7
So clearly, the terms are 7a, 5a, -21, and 7. The like terms are (7a, 5a) and (-21, 7).
Expression 2: Consider the following expression: \(7a^{2}+5a-21a+7a^{2}\)
We have to identify the terms and the like terms in this expression. If you notice there are exponents expressed as well.
Answer: The terms in the given expression are \(7a^{2}, 5a, -21a\text{ and }7a^{2}\). The like terms are (\(7a^{2}, 7a^{2}\)) and ( \(5a, -21a\)).
A linear expression is an algebraic expression in which the exponent of each variable is one.
The table below shows linear and nonlinear expressions:
Example 1: Simplify the given algebraic expressions.
(a) \(6a-4a\)
(b) \(6.7b-2.2b+b\)
(c) \(-3a-6a+2b+5b\)
(d) \((\frac{3}{4})a+10-(\frac{1}{2})a-2\)
Solution:
Part (a)
We have: \(6a-4a\)
\(6a-4a~=~(6-4)~a~\) Using the distributive property
= \(2a\)
Part (b)
We have: \(6.7b-2.2b+b\)
= \(6.7b-2.2b+(b)\) Using the multiplication property of 1
= \((6.7-2.2+1)\times b\) Using the distributive property
= \(5.5b\)
Part (c)
We have: \(-3a-6a+2b+5b\)
= \((-3~-6)a~+~(2~+~5)b\) Using the distributive property
= \(-9a+7b\)
Part (d)
We have: \((\frac{3}{4})a+10-(\frac{1}{2})a-2\)
= \((\frac{3}{4})~a~-~(\frac{1}{2})~a~+~10~-~2\) Using the commutative property of addition
= \([(\frac{3}{4})-(\frac{1}{2})]a~+~10~-~2\) Using the distributive property
= \((\frac{1}{4})a+8\)
Example 2: A match ticket and a drink are purchased by each member of a group. If the group comprises six people, how much does the group pay?
Solution: Write an expression:
The tickets and the drinks are bought in equal numbers. So, let a denote both the number of tickets and the number of drinks.
\(\text{cost of a ticket }\times \text{number of tickets + cost of a drink}\times \text{ number of drinks}\)
= 5a + 2a
= (5 + 2)a Using the distributive property
= 7a
The expense for each person to watch a match is $7. To find the expense for 6 people, we will have to replace “a” with 6.
So, for 6 persons = 7 × 6
= 42
The total cost for 6 people will be $42
Example 3: Solve the given linear expressions using different operations.
(a) \((a~-~5)~+~(4a~+~6)\)
(b) \((-3a~+~2)~+~(10a~+~4)\)
(c) \((2a~+~7)~-~(-a~+~3)\)
(d) \((-5a~+~1)~-~~(8a~+~4)\)
Solutions:
Part (a)
We have: \((a~-~5)~+~(4a~+~6)\)
\(a-5\)
\(4a+6\)
\({+}\) _____
\(5a+1\) Using the vertical addition method
The sum is \(5a +1\)
Part (b)
We have: \((-3a~+~2)~+~(10a~+~4)\)
\(=~ -3a~+~2~+~10a~+~4\) Rewrite the given equation
\(=~ -3a~+~10a~+~2~+~4\) Using the commutative property of addition
\(= ~(-3~+~10)~\times~ a~+~(2~+~4)\) Group the like terms
\(~=~ 7a~+~6\) Combine the like terms
The sum is \(7a~ +~6\)
Part (c)
We have: \((2a~+~7)~-~(-a~+~3)\)
\( = (2a~+~7)~+~a~-~3\)
\(2a+7\)
\(a-3\)
\({+}\) _____
\(3a+4\) Using the vertical addition method
The difference is \(3a~ +~4\)
Part (d)
Given that: \((-5a~+~1)~-~(8a~+~4)\)
\( =~ -5a~+~1~-~8a~-4\) Rewrite the given equation
\( =~ -5a~-8a~+~1~-4~\) Using the commutative property of addition
\( = (-~5~-8)~\times ~a~+~(1~-~4)\) Group the like terms
\(=~ -13a~+~5\) Combine the like terms
The difference is \(-13a~-3\)
Example 4: A baseball kit costs 3a dollars. You sell each kit for \(6a-~2\) dollars after it has been assembled. Calculate your profit for each baseball kit that you sold if each kit costs 2$ additional for assembling.
Solution:
The information on how to buy and sell baseball kits is given. We need to calculate the profit.
Calculate the difference between the expressions for the selling and buying prices. The expression should then be simplified and interpreted.
Assembling cost is 2$ so the total cost price is \((3a~+~2)\)$.
Profit = Selling price – Purchase Price
= (6a – 2) – (3a + 2)
= 6a – 3a – 2 – 2 Group like terms and simplify
= 3a – 4
You will get a profit of 3a – 4 dollars from each baseball kit that you have sold.
A two-variable linear equation is defined as a linear relationship between x and y, or two variables in which the value of one (usually y) is dependent on the value of the other (usually x).
Because x is the independent variable in this case, and y is dependent on it, y is referred to as the dependent variable.
Algebraic expressions fall under three categories:
- Monomial Expressions
- Binomial Expressions
- Polynomial Expressions