7.03 Solving equations with algebra

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$4 equals $3$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.

Solving simple equations

Before we take a step forward to using algebra for solving equations, let's look back at what solving an equation requires.

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$3+x=10 has the solution $x=7$x=7. We didn't use any particular method here, we just knew that $3+7$3+7 is equal to $10$10, so the number replacing the pronumeral must be $7$7.

When solving equations like this, we are effectively reducing the equation to a 'fill in the blank' problem.

While this approach to solving equations is very easy and straightforward, it is only useful when the calculations are easy enough to solve just by looking at them. For more complex equations, we will need a method of simplifying the equation to one that we can solve more easily.

Isolating the pronumeral

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.

Notice that when we applied the subtraction of $8$8 to the equation, we applied it to both sides of the equation. This is very important!

Reverse operations

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.

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.

Reflect: When finding the reverse operation, the operation will always be the reverse of that in the original operation while the number stays the same.

Remember that terms like $4n$4n represent the multiplication $n\times4$n×4 and fractions like $\frac{m}{6}$m6​ represent the division $m\div6$m÷​6.

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 $\frac{x}{7}=12$x7​=12.

We can write this using the basic operations as $x\div7=12$x÷​7=12

To isolate $x$x, we want to reverse the division by $7$7. Since multiplication reverses division, the reverse operation will be 'multiply by $7$7'. Applying this to both sides of the equation gives us:

$x\div7\times7=12\times7$x÷​7×7=12×7

After cancelling out the division and multiplication of $7$7, we get:

$x=12\times7$x=12×7

Notice that when we applied the reverse operation, it had the same effect as turning division on the left-hand side into multiplication on the right-hand side.

Swapping the side and sign of the operation reverses it.

This shows us that, by changing both the side and sign of an operation, we can reverse the operation in a single step.