Middle Years

# 6.04 Three step equations

Lesson

When trying to solve complicated problems, it is best to take them one step at a time. This applies to equations as well.

We learned in the previous lesson that we can isolate the pronumeral in an equation by reversing the operations applied to it. When solving three step equations, we can use the same ideas to solve these complicated equations one step at a time.

### Which step to solve first?

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=\frac{3x+8}{4}$11=3x+84 we can see that the expression containing the pronumeral was formed by applying the following operations:

1. Multiply by $3$3
2. Add $8$8
3. Divide by $4$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$4
2. Subtract $8$8
3. Divide by $3$3

Applying these reverse operations gives us:

 $11$11 $=$= $\frac{3x+8}{4}$3x+84​ $44$44 $=$= $3x+8$3x+8 Reverse the division $36$36 $=$= $3x$3x Reverse the addition $12$12 $=$= $x$x Reverse the multiplication

Following these steps isolates the pronumeral and solves the equation.

### The order of reverse operations

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:

2. Then multiplication and division outside any brackets
3. Repeat steps 1 and 2 for expressions inside brackets

#### Worked Example

Solve the equation: $-3\left(x-14\right)+5=29$3(x14)+5=29

Think: The order for reverse operations, we want to reverse the addition, then the multiplication, then the subtraction.

Do: Since addition and subtraction reverse each other and division reverses multiplication, we can solve the equation like so:

 $-3\left(x-14\right)+5$−3(x−14)+5 $=$= $29$29 $-3\left(x-14\right)$−3(x−14) $=$= $24$24 Reverse the addition $x-14$x−14 $=$= $-8$−8 Reverse the multiplication $x$x $=$= $6$6 Reverse the subtraction

#### Practice questions

##### Question 1

Solve the equation $3\left(4s+1\right)=-21$3(4s+1)=21.

1. Enter each line of working as an equation.

##### Question 2

Solve the equation $-\frac{u}{4}+15=8$u4+15=8.

1. Enter each line of working as an equation.

##### Question 3

Solve the equations $\frac{8x+4}{5}=-4$8x+45=4.

1. Enter each line of working as an equation.