An equation is a mathematical sentence stating that two expressions are equal. That is, they have the same value. We can think of it as being balanced.

Here are some examples:

$3x+2=4$3x+2=4

$4.25=3\frac{1}{2}+n$4.25=312+n

$\frac{y}{3}=2y+5$y3=2y+5

Equations often contain letters or symbols are used to represent an unknown quantity. These symbols are called variables. As seen above, variables can appear on one or both sides of an equation.

An equation is like a sentence. We would like to translate a written sentence to a mathematical sentence. Where we see "is" or "equals", we will put an equals sign.

Worked example

Question 1

Adam thought of a number, doubled it and added $5$5 to get $13$13. Write an equation that represents this scenario.

Think: There is an unknown quantity, "a number", so we should use a variable to represent it. Let's use $n$n.

Do:

$n\times2+5$`n`×2+5	$=$=	$13$13	We take a number, double it and then add $5$5, this is equal to $13$13
$2n+5$2`n`+5	$=$=	$13$13	We prefer to write a product of a number and variable with the number first

Question 2

What scenario could the equation $3a=2b+1$3a=2b+1 be representing?

Think: There is certainly not a single correct answer here, so here is one possible scenario.

We should start by stating what $a$a and $b$b represent.

Do: Let $a$a be the number of apples and $b$b be the number of bananas. The cost of $3$3 apples is $1$1 dollar more than the cost of $2$2 bananas.

Reflect: How many different scenarios can you come up with? What do all the scenarios have in common? What is different?

Practice questions

Question 3

Is $5r-15=0$5r−15=0 an expression or an equation?

It is an expression.
A
It is an equation.
B

Question 4

A number (call it $n$n) plus $four$ is equal to $seven$ .

Write the equation using mathematical symbols.
What is the value of $n$n?

Question 5

$Roxanne$ has been out picking flowers, and has $40$40 in total. When she returns, she puts them in $5$5 different vases.

If she puts $p$p flowers in each vase, rewrite the following sentence using algebra:

"There are $5$5 groups of $p$p flowers, which make $40$40 in total."

Identifying solutions

We say that a value for a variable is a solution to an equation if we can substitute it into the equation and it makes the number sentence true.

For example, if we wanted to find the solution to the equation $x+1=3$x+1=3, we want to find a value for $x$x that makes that equation true. This statement is true when $x=2$x=2 because $2+1=3$2+1=3.

When we're figuring out whether a value is a solution, we need to see whether the left-hand side of the equation is the same as the right-hand side. We can think of it as a see-saw.

For example, if we have the equation $x+12=20$x+12=20, we could think of it visually as:

If $12$12 was removed from the left-hand side of the seesaw, it would look unbalanced like this:

So how do we balance the equation again? We need to remove $12$12 from the right-hand side as well:

So $x=8$x=8 satisfies the equation $x+12=20$x+12=20 because $8+12=20$8+12=20.