Introduction
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. This equation joins the two expressions, 3\times5 and 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. This equation also joins two expressions, 3\times5 and 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.
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.
Examples
Example 1
Is the following equation true or false?\dfrac{96}{8} = 106-94
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.
Make 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.
A teacher wants to divide their class of 30 students into groups of 5. How many groups will they need?
We can set this problem up using an equation.
Since the teacher wants to divide 30 by some number to get 30, the equation will look like this:\dfrac{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, since \dfrac{30}{6}=5.
But what if we make the equation false?
We can make the equation false by filling the blank space with the number 3. This would give us \\ \dfrac{30}{3}=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.
Examples
Example 2
What value for s will make the following equation true?s-7=14Write your answer in the form s=⬚.
The number that makes an equation true is called the solution.
The unknown in an equation can be represented by a pronumeral like x,\, J, or s.
Convert 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.
Examples
Example 3
Which of the following equations matches the description 'the product of eight and s is taken away from sixty six. The result is equal to ten'?
We can describe equations with words quite easily by translating all the mathematical symbols into words.