Introduction
While we can solve simple equations by reversing a single operation or just by observing a solution, more complicated equations require a more methodical method - algebra.
Two step equations
When solving more difficult equations, we want to use algebra to isolate the pronumeral by reversing the operations applied to it.
This is similar to working backwards, where we reversed the operations one at a time to isolate the pronumeral except, instead of observing the value for simpler expressions, we are directly reversing the operations.
Consider the equation, 4x+7=31.
To solve this equation with algebra, we notice that the expression on the left-hand side of the equation has been built by applying the operations 'multiply by 4' and 'add 7' to the pronumeral.
To reverse these operations, we can apply the reverse operations 'subtract 7' and 'divide by 4' to both sides of the equation.
By applying these operations, we isolate x and find the solution:
We knew that reversing the operations that were used to build the expression would isolate that pronumeral, but how did we know what order to apply them in?
One way to know what order to apply the reverse operations in is to consider the order in which the operations were applied to build the expression. This approach is explained in more detail when looking at breaking down expressions when using non-algebraic methods for solving two-step equations.
However, a faster way to find the solution is to apply the reverse operations according to the order of reverse operations.
What does this mean?
If we take another look at the equation 4x+7=31, we can see that we should reverse the addition first since that will simplify the left-hand side to a single term, while reversing the multiplication first would result in expressions containing fractions which does not simplify the expression.
It is for this reason that we usually apply the reverse operations according to the steps:
Start with addition and subtraction
Then multiplication and division
Repeat steps 1 and 2 for expressions inside brackets
We can see this in action with a more complicated equation, 7(y+6)-8=55.
We can see that the operations used to build the expression on the left-hand side were 'add 6', 'multiply by 7', and 'subtract 8'.
This means that the reverse operations we want to apply to the equation will be 'subtract 6', 'divide by 7', and 'add 8'.
We can then apply these in the order of reverse operations:
Add 8
Divide by 7
Subtract 6
Notice that we apply the reverse operation 'subtract 6' last since y+6 is contained within a pair of brackets.
Applying these operations isolates y on the left-hand side of the equation to give us the solution: y=3
Now that we know what operations to apply and in what order, we can solve equations quickly and accurately using algebra.
Examples
Example 1
Consider the equation 5(n-15)=35.
If we want to make n the subject of the equation, what is the first step we should take?
What is the second step we should take to make n the subject?
Apply the steps found in parts (a) and (b) to find the solution to the equation.
Example 2
Solve the equation \dfrac{u+7}{2}=5.
When applying reverse operations to isolate a pronumeral, we apply them according to the order of reverse operations:
Addition and subtraction
Multiplication and division
Expressions inside brackets
Notice that this order is the reverse of the usual order of operations.
When writing an equation, swapping the left and right-hand sides of the equation does not change the solution.
Move the pronumeral first
With the skills we now have, we can reverse as many operations as we need to isolate our pronumeral. But what if we want to solve equations like 4-x=17 or \dfrac{5}{x}=20?
In both of these cases we can see that the pronumeral is actually part of the operation being applied, so we can't isolate it so easily.
However, if we reverse the operation containing the pronumeral, we can move it so that the equation can be solved using the skills we just learned.
Examples
Example 3
Solve the equation \dfrac{-44}{p}=11.
If the pronumeral is actually part of the operation being applied, we can reverse the operation containing the pronumeral.
For example, in the following equation we are dividing by x: \dfrac{5}{x}=2 so we can use the reverse operation and multiply both sides by x: \begin{aligned} \dfrac{5}{x}\times x & =2 \times x \\ 5 &=2x \end{aligned}