7.02 Building algebraic expressions

When writing a numeric expression we use numbers and basic operations to build up a number sentence that can be later calculated. Algebraic expressions are the same as numeric expressions except that they also use some new algebraic tools. These new algebraic tools are variables and coefficients.

 

Variables

We use algebraic expressions when we want to write a number sentence but don't know all the numbers involved.

For example: What is the total weight of a cat and a dog?

In this example we know that the total weight will be the weight of the cat added to the weight of the dog, but we don't know the number for either of these.

What we do instead is pretend that we know what these numbers are and replace them with variables.
In this case, let's use $c$c for the weight of the cat and $d$d for the weight of the dog.

Now we can write the number sentence as:

Total weight = $c+d$c+d

 

This is an algebraic expression as it is a number sentence that uses variables in the place of some numbers.

 

 

Let's try converting a word problem into an algebraic expression with the help of some variables.

 

Worked example

There are some red fish and some blue fish in a tank. If $5$5 yellow fish are added to the tank, how many fish are now in the tank?

Think: We don't know how many red fish or blue fish there are in the tank so instead we can use variables to represent their numbers.

Do: Let's use $r$r for the red fish and $b$b for the blue fish. This means that:

• Number of red fish = $r$r
• Number of blue fish = $b$b
• Number of yellow fish = $5$5

Using these variables we can write an algebraic expression for the total number of fish in the tank:

Total number of fish = $r+b+5$r+b+5

Reflect: Since we didn't have numeric values for the number of red or blue fish we simply replaced them with variables. We then wrote the algebraic expression for the total number of fish as a sum using these variables. Notice that we couldn't evaluate $r+b$r+b as they are different variables with unknown values.

It should be noted that the choice of $r$r and $b$b as the variables is arbitrary and we can use whatever symbol we want for our variables, provided they don't already represent a value.

 

 

Coefficients

Coefficients are used in algebraic expressions to represent how many sets of a variable we have. They are written in front of a variable without a multiplication symbol like so:

 

Notice how we don't need the multiplication symbol to represent multiple sets of a variable.
This is because there is no danger of mixing up a coefficient next to a variable with any other term, whereas if we did this for numbers they would get mixed up with two digit numbers.

 

 

Coefficients are a bit different from multiplication though, since they also include the sign of the term.
This means that a negative term, $-6q$−6q for example, has a coefficient of $-6$−6.

We can see this more clearly in a longer expression.

Consider the expression: $4x-3y+7z$4x−3y+7z

By breaking up the expression into its individual terms we can determine the coefficients of each variable.

 

Term	Coefficient	Variable
$4x$4x	$4$4	$x$x
$-3y$−3y	$-3$−3	$y$y
$+7z$+7z	$7$7	$z$z

 

From this we can see that the coefficient of $y$y is $-3$−3, since it is a negative term, and the coefficient of $z$z is $7$7, since it is a positive term. If there is no sign in front of a term we assume that the term is positive, so we know that the coefficient of $x$x is $4$4.

We can also have algebraic terms where the coefficient is a fraction.
Consider: $v\div4$v÷​4$=$=$\frac{v}{4}$v4​$=$=$\frac{1}{4}\times v$14​×v.
Since dividing by a number is the same as multiplying by its reciprocal, dividing by $4$4 gives us a coefficient of $\frac{1}{4}$14​.

What about variables that don't appear to have coefficients?
Consider the term $x$x.
Since $x$x is equal to $1\times x$1×x which is also equal to $1x$1x, it actually has a coefficient of $1$1.
Whenever a variable has no written coefficient, its coefficient can be assumed to be $1$1.

Similarly, the coefficient of $-x$−x is $-1$−1.

We can apply our new understanding of coefficients to questions like this:

 

Practice question

Question 1

 

Basic operations in algebra

Aside from the use of coefficients in multiplication, the basic operations work almost the same for algebraic terms as they do for numbers.

Between variables and numbers we have:

 

Word Expression	Algebraic Expression	Simplified Algebraic Expression
three more than $x$x	$x+3$x+3	$x+3$x+3
three less than $x$x	$x-3$x−3	$x-3$x−3
the quotient of $x$x and three	$x\div3$x÷​3	$\frac{x}{3}$x3​
the product of $x$x and three	$x\times3$x×3	$3x$3x

 

Between variables and other variables we have:

 

Word Expression	Algebraic Expression	Simplified Algebraic Expression
$y$y more than $x$x	$x+y$x+y	$x+y$x+y
$y$y less than $x$x	$x-y$x−y	$x-y$x−y
the quotient of $x$x and $y$y	$x\div y$x÷​y	$\frac{x}{y}$xy​
the product of $x$x and $y$y	$x\times y$x×y	$xy$xy

 

As we can see from the tables, addition and subtraction in algebraic expressions does not usually simplify.
The only time they will simplify is when they  are like terms.

It should also be noted that the division doesn't actually simplify but is instead written as a fraction, which is slightly more compact and removes the need to use brackets in more complicated expressions, for example $4\div\left(x+3\right)=\frac{4}{x+3}$4÷​(x+3)=4x+3​.

Technically, the multiplication between a variable and a number also uses a more compact form by removing the multiplication symbol, but in this case using a coefficient is considered simplifying.

The same can be said for the multiplication between different variables except, in this case, there is no coefficient and instead we have two variables.

These operations will work the same way when applying more than one of them.

 

Worked example

Write the algebraic expression for "$q$q less than the product of $4$4 and $p$p".

Think: The operations involved are subtraction (denoted by "less than") and multiplication (denoted by "the product of"). Following the order of operations we should write the product first and then use it in the subtraction.

Do: We can simplify the multiplication between a number and a variable by writing the number as the coefficient of the variable. We can do this for "the product of $4$4 and $p$p", writing it as $4p$4p.

We can then include this in the subtraction operation so that the word expression is now "$q$q less than $4p$4p", which is $4p-q$4p−q.

Reflect: We converted the word expression into an algebraic expression by converting one operation at a time, following the order of operations.

Each time we convert a word operation into an algebraic operation we can replace that operation with its algebraic term or expression.
This works because, unless we have like terms, the operations do not affect each other.

Try applying this new knowledge to the questions below.

 

Practice questions

Question 2

Question 3