9.06 Three step equations

We know that we want to reverse the operations to isolate the pronumeral, the question is which operation should we should reverse first?

When isolating the pronumeral we want to reverse the operations in the opposite order to which they were applied.

For example, in the equation 11=\dfrac{3x+8}{4} we can see that the expression containing the pronumeral was formed by applying the following operations:

1. Multiply by 3

2. Add 8

3. Divide by 4

In order to isolate the pronumeral, we want to reverse these operations starting from the last applied to the first. As such, the reverse operations we should apply are:

1. Multiply by 4

2. Subtract 8

3. Divide by 3

Applying these reverse operations gives us:

Following these steps isolates the pronumeral and solves the equation.

Notice that, in the example above, we reversed the division, then the addition and finally the multiplication. But this doesn't match our usual order of operations at all. We used this order because we also need to pay attention to the position of brackets (and the numerator of fractions) when solving equations.

If part of the expression is enclosed in a pair of brackets (or in the numerator) it means that some operation was applied to everything inside those brackets and we will need to reverse that operation first. It is for this reason that we reversed the division first in the example above.

Knowing this, we can reverse the operations applied to the pronumeral according to the order:

1. Start with addition and subtraction outside any brackets

2. Then multiplication and division outside any brackets

3. Repeat steps 1 and 2 for expressions inside brackets

Examples