In this explainer, we will learn how to read and write algebraic expressions, model them, and apply this to real-life situations.
In mathematics we are often looking for relationships between objects and looking for unknown values. To help represent an unknown value, we need to introduce some notation. We usually use letters to represent these unknown values, often or . These are called variables since their value can vary based on the relationship. It is also worth noting that these are sometimes called indeterminates since their value may not be determined. When writing algebraic expressions, we often refer to numbers like 5 as constants since their value remains fixed.
It is important to note that variables behave exactly like normal numbers. We can add and subtract variables and multiply and divide by variables.
For example, imagine we are told that is a number and we want to write an expression to represent the value of increased by 10. We can do this by just adding 10 to , giving us the expression . We follow this process for any operation we can apply to a number.
Another operation we can use with variables is exponentiation. If, for example, we are told that is a number, then we can represent the cube of as .
However, we do need to be careful when applying operations with variables since these variables can take on multiple values. For example, we need to be cautious not to divide by 0.
In our first example, we will write an expression to represent a simple relationship.
Example 1: Writing an Algebraic Expression to Represent a Given Relationship
Write an expression for a number decreased by 75.
Since the variable needs to be decreased by 75, we need to subtract 75 from . We write that as the expression
This expression represents a value that is 75 less than .
When working with variables, the multiplication operations behave as we would expect. There are, however, some important differences in notation that you may encounter. When writing an expression in which a constant is multiplied by a variable, we often omit the multiplication symbol and write the constant first. For example, if we wanted to represent the product of and 3, we may want to write this is as . However, we simplify this expression by removing the product operator and writing the constant first, giving us .
We can also do something similar when multiplying variables: we can remove the multiplication operator, and we usually write the variables in alphabetical order. Hence, we often write the product of the variables and as . Of course, since multiplication is commutative, we could equally express this as .
Thus far, we have only considered a single operation. However, we will often need to apply a series of operations to one or more variables. This comes with the added difficulty of getting the order of operations correct. For example, if we need to represent an unknown number (variable) that is multiplied by 2 and then increased by 1, we need to make sure that multiplication by 2 is carried out first. We can do this by recalling the order of operations and properly using mathematical notation.
Definition: Order of Operations
We can recall that the order in which operations are carried out by using the acronym PEMDAS. The order is as follows:
This then allows us to find an expression representing multiplied by 2 and then increased by 1. We see that multiplication appears before addition in the order of operations, so we can write for multiplied by 2 and then to add 1 to this value.
If we had instead wanted to write increased by 1 and then multiplied by 2, we would need to note that, since addition is below multiplication in the order of operations, we will need to use parentheses. We do this in order to ensure that the addition operation is carried out before the multiplication operation. We first write to represent increased by 1 and then we need to multiply this by 2, giving us . We can then see that the addition is inside the parentheses and so is carried out first, and then we multiply by 2.
It is important to note that these two operations do not result in identical expressions. This can be shown by expanding the parentheses:
Note that the expression above is not the same as . In order to avoid errors, we should take care to correctly interpret the order of operations specified by a word problem.
Let’s now see an example where we need to write an expression for the age of a person.
Example 2: Writing an Algebraic Expression to Represent a Given Relationship
Given that Ramy is years old now, write an expression for his age 5 years ago.
If Ramy’s age is , then his age 5 years ago will be 5 less than . Thus, we need to subtract 5 from . We recall that the order of operations tells us that multiplication comes before subtraction, so we can subtract 5 from the expression directly to get .
Hence, Ramy’s age 5 years ago is given by .
Let’s now see an example where we need to write an expression to represent a series of operations applied to a given variable.
Example 3: Writing an Algebraic Expression to Represent a Given Relationship Involving an Exponent
Express the following in algebraic form: the square of a number that is then multiplied by 19.
In this question, we want to represent a series of operations applied to a given variable. We want to represent the variable that is first squared and then multiplied by 19. It is important we get the order of operations correct; otherwise, we might write multiplied by 19 and then squared.
We first want to square ; we can write this as . We then want to multiply this by 19; we could write this using parentheses as . This is because we know that the operation inside the parentheses is carried out first.
It turns out that we do not need parentheses since exponentiation is already carried out before multiplication according to the order of operations, so the parentheses are not necessary.
Hence, we can write the answer as .
In our next example, we will determine an expression for the length of a line given expressions for the lengths of two segments that make up the line.
Example 4: Writing Algebraic Expressions That Represent the Length of a Drawn Line
Write an expression that represents the length of in the given figure.
We first note that the length of a line segment is given by the sum of the mutually exclusive line segments that make up the entire line segment. Let’s call the following point .
We can see that . Since and , we can add these expressions together to find an expression for .
This gives us .
In our next example, we will find an algebraic expression for the perimeter of a rectangle using a given diagram.
Example 5: Using Algebraic Expressions to Describe the Perimeter of a Triangle
Write an algebraic expression for the perimeter of the triangle below.
We start by recalling that the perimeter of a triangle is the sum of all of its side lengths. We note that the base of the triangle has a length that is the sum of and , written . We can add all of these side lengths together to get
Since we can add these values in any order, we can remove the parentheses to get
Hence, the perimeter of the triangle is .
It is worth noting that addition is commutative, so we can add the terms in the previous answer in any order to get an equivalent answer. For example, we could also have given the answer or any other variation of the terms. However, we usually give the terms with the variables in alphabetical order with the constant at the end.
In our next example, we will find an algebraic expression representing a series of operations involving two variables.
Example 6: Writing an Algebraic Expression to Represent a Series of Operations Involving Two Variables
Write an expression for the product of the cube of and the difference between a number and 9, all divided by 2.
We want to represent a series of operations applied to multiple variables. That is, the product of the cube of and the difference between a number and 9, all divided by 2. It is important that we get the order of operations correct, and to do this, we need to consider the wording of the operations.
Let’s start by determining each of the factors in this product. The first factor is the cube of , written as , and the second factor is difference between and 9, written as . Therefore, the product of these expressions is written as .
Let’s now take note of the last few words of the question. After describing an expression, the one we have just formed, the question then tells us that this is “all divided by 2.” This signifies that division by 2 is the final operation that must be applied.
Hence, we then divide this product by 2 to get
In our final example, we will find an algebraic expression representing a mathematical description involving two variables.
Example 7: Writing an Algebraic Expression to Represent a Mathematical Description Involving Two Variables
Write an expression for double the square of , which is then divided by 9 less than the cube of .
In this question, we want to represent a series of operations applied to multiple given variables. We want to represent double the variable squared and then divided by 9 less than the cube of . It is important that we get the order of operations correct.
Let’s start with the first part of the relationship, representing double the variable squared. We need to first square the variable , which we can write as . We then need to double the expression. We note that, in the order of operations, exponentiation is carried out before multiplication, so we can write this as to say that is squared and then multiplied by 2.
Let’s now consider the second part of the relationship, dividing this expression by 9 less than the cube of . Let’s determine an expression for the divisor. We need to cube , written as , and then we need to decrease this value by 9. Since subtraction is carried out after exponentiation, the divisor can be written as .
Finally, we need to divide the first expression by the second expression, this gives us
It is worth noting that, when writing fractions in this form, we usually remove the parentheses to make the expression look simpler. This does not break our order of operations since it is the convention to evaluate the numerator and the denominator of the quotient first before carrying out the division operation.
This gives us
Let’s finish by recapping some of the important points from this explainer.
- We can represent unknown numbers as letters called variables or indeterminates.
- Since variables are numbers, we can apply operations to these variables to represent relationships.
- When multiplying a constant and a variable, it is the convention that we write the constant first and remove the multiplication operator. This is the standard notation for multiplication.
- The order of operations is often important to make sure that the relationship is carried out in the correct order.