Introduction
While guessing and checking to find the solution to an equation does work, it isn't always the fastest method. Sometimes, we can interpret the equation as a simple calculation such as 'what number divided by 4 equals 3?'. But for harder questions involving multiple operations the answer will not always be obvious. In such cases, we can solve an equation using algebra.
Simple equations
In order to solve an equation containing a pronumeral, we want to find a value that replaces the pronumeral to make the equation true. While there are multiple methods available to us for solving equations in a systematic way, there are also times where we can just look at the equation and solve it.
For example, we can see quite easily that the equation 3+x=10 has the solution x=7. We didn't use any particular method here, we just knew that 3+7 is equal to 10, so the number replacing the pronumeral must be 7.
When solving equations like this, we are effectively reducing the equation to a 'fill in the blank' problem.
When trying to find the value for a pronumeral that will solve the equation, we can avoid the guess and check method by isolating the pronumeral on one side of the equation and directly solving for its value. This is also referred to as "making the pronumeral the subject of the equation".
But how can we isolate the pronumeral?
In order to isolate the pronumeral, we want to apply operations to the equation so that the pronumeral ends up alone on one side of the equation. In other words, we make it the subject of the equation.
When isolating the pronumeral, we need to know which operation to apply so that the pronumeral will be on its own at the end. The fastest way to do this is to reverse any operations that are currently being applied to the pronumeral.
To do this, we need to know which operations are reversed by which.
| Operation | Reverse operation | Example |
|---|---|---|
| \text{Addition} | \text{Subtraction} | x+4-4=x |
| \text{Subtraction} | \text{Addition} | x-3+3=x |
| \text{Multiplication} | \text{Division} | y\times4\div4=y |
| \text{Division} | \text{Multiplication} | y\div 2\times 2=y |
We can see that four basic operations can be sorted into pairs,
Addition and subtraction reverse each other
Multiplication and division reverse each other
As we can see in the table, when operations that reverse each other are applied to the same pronumeral (or number) they will cancel out.
Examples
Example 1
Solve the equation v-4=44.
The four basic operations can be sorted into pairs,
Addition and subtraction reverse each other
Multiplication and division reverse each other
When operations that reverse each other are applied to the same pronumeral (or number) they will cancel out.
Single step equations
When we reverse an operation in an equation, we are applying the reverse operation to both sides of the equation.
Consider the equation \dfrac{x}{7}=12.
We can write this using the basic operations as x\div7=12.
To isolate x, we want to reverse the division by 7. Since multiplication reverses division, the reverse operation will be 'multiply by 7'. Applying this to both sides of the equation gives us:x\div7\times7=12\times7After cancelling out the division and multiplication of 7, we get:x=12\times7
This shows us that, by changing both the side and sign of an operation, we can reverse the operation in a single step.
Examples
Example 2
Solve the equation 24+p=59.
When we reverse an operation in an equation, we are applying the reverse operation to both sides of the equation.