7.01 Introduction to equations
One of the most common ways in which we express mathematical relationships is by using an equation. An equation joins two expressions together using an equals sign to make a number sentence that proposes that the two expressions are equal.
One example of an equation is $3\times5=15$3×5=15. This equation joins the two expressions, $3\times5$3×5 and $15$15, using an equals sign to show that they are equal. This is an example of a true equation.
Another example of an equation is $3\times5=16$3×5=16. This equation also joins two expressions, $3\times5$3×5 and $16$16, using an equals sign to show that they are equal. However, because the two expressions are not equal in value, this is an example of a false equation.
An equation is a number sentence that joins two expressions with an equals sign. An equation is true if the expressions on either side of the equals sign are equal in value, otherwise the equation is false.
True or false?
We now know that an equation is true if the expression on the left-hand side of the equals sign is equal to the expression on the right-hand side. We can check if this is the case by evaluating both expressions and seeing if they are equal in value or not.
Worked example
Example 1
Is the following equation true or false?
$20+6=7\times4$20+6=7×4
Think: In order to check if the equation is true or false, we first need to evaluate the expressions on either side of the equals sign.
Do: We can evaluate the expression $20+6$20+6 by performing the addition.
$20+6=26$20+6=26
We can evaluate the expression $7\times4$7×4 by performing the multiplication.
$7\times4=28$7×4=28
Since we have found that the expressions are not equal in value, we can say that the equation is false.
Reflect: To check if the equation was true or not, we evaluated the expressions on either side of the equals sign and compared their values. Since the values of the expressions were not equal, the equation was false.
We can use this method to solve questions like this.
Practice question
Question 1
Is the following equation true or false?
$\frac{96}{8}=106-94$968=106−94
True
AFalse
B
Making equations true
When applying equations to the real world, we find that we usually want to make the equations true because we are trying to solve problems. We don't want to make the equations false because that won't solve our problems.
Consider the following scenario.
Exploration
A teacher wants to divide their class of $30$30 students into groups of $5$5. How many groups will they need?
We can set this problem up using an equation.
Since the teacher wants to divide $30$30 by some number to get $5$5, the equation will look like this:
$\frac{30}{\editable{}}=5$30=5
The blank space represents the number of groups that the teacher wants to find.
In order to make the equation true, the blank space should be filled with the number $6$6, since $\frac{30}{6}$306 is equal in value to $5$5.
But what if we make the equation false?
We can make the equation false by filling the blank space with the number $3$3. This would give us $\frac{30}{3}=5$303=5 which is not true. Since this answer does not solve the teacher's problem, we do not find this equation very useful for our problem.
Since making the equation true is what solves the problem, we call the number that makes an equation true a solution.
Worked example
Example 2
In $5$5 years from now, Jason will be $12$12 years old. How old is he now?
Think: We can find his current age by setting up an equation and then finding the solution.
Do: We know that Jason's current age plus $5$5 will be equal to $12$12. We can express this as the equation:
$\editable{}+5=12$+5=12
What number can we replace the blank with to make the equation true?
Since $7+5=12$7+5=12 is true, the solution to this equation is $7$7.
Therefore, Jason is currently $7$7 years old.
Reflect: We can solve real world problems by converting them into equations and then finding values that make those equation true.
In this example we used a blank to represent Jason's current age. However, we could also have used the pronumerals $x$x or $J$J.
Since the equations $x+5=12$x+5=12 and $J+5=12$J+5=12 are both true when the letter is equal to $7$7, we can see that the choice of pronumeral doesn't affect the solution.
For some practise on making equations true, try the problem below.
Practice question
Question 2
What value for $s$s will make the following equation true?
$s-7=14$s−7=14
Write your answer in the form $s=\editable{}$s=.
Converting between words and equations
It is important when working with equations that we can understand them. It is even more important that we can convert word problems into equations so we can solve them.
We can describe equations with words quite easily by translating all the mathematical symbols into words.
Worked examples
Example 3
Describe the equation $\frac{45}{x}=3$45x=3 using words.
Think: We can convert this equation into a word description by translating each mathematical symbol into words. We should also remember that $\frac{45}{x}$45x is the same as $45\div x$45÷x.
Do: We can convert each mathematical symbol into words like so:
| $45$45 | $\div$÷ | $x$x | $=$= | $3$3 |
|---|---|---|---|---|
| forty five | divided by | $x$x | is equal to | three |
So a description for the equation $\frac{45}{x}=3$45x=3 can be:
forty five divided by $x$x is equal to three
Reflect: We can describe equations by translating mathematical symbols into words.
It should be noted that there is more than one way to describe an equation. In this case, the description could also be:
the quotient of forty five and $x$x is equal to three
When converting from word descriptions into equations we can use the same method, but in reverse.
Example 4
Write an equation that matches the description "the product of eight and $q$q is equal to $p$p less than twenty one".
Think: Before we translate our words into mathematical symbols, we can make the description simpler by remembering that "product" implies multiplication and "less than" implies subtraction.
Do: We can make the description simpler by replacing "the product of eight and $q$q" with "eight times $q$q" and replacing "$p$p less than twenty one" with "twenty one minus $p$p".
We can then convert the words directly into mathematical symbols, like so:
| eight | times | $q$q | is equal to | twenty one | minus | $p$p |
|---|---|---|---|---|---|---|
| $8$8 | $\times$× | $p$p | $=$= | $21$21 | $-$− | $p$p |
After removing any unnecessary multiplication symbols we get the equation:
$8q=21-p$8q=21−p
Reflect: We can convert descriptions into equations by simplifying the language and then directly translating into mathematical symbols.
We can practise converting from descriptions to equations with questions like this.
Practice question
Question 3
Which of the following equations matches the description "the product of eight and $s$s is taken away from sixty six. The result is equal to ten"?
$8\left(66-s\right)=10$8(66−s)=10
A$66-8s=10$66−8s=10
B$8s-66s=10$8s−66s=10
C$8s-66=10$8s−66=10
D