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Year 8

5.02 Three step equations

Lesson

Introduction

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.

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:

\displaystyle 11\displaystyle =\displaystyle \dfrac{3x+8}{4}
\displaystyle 44\displaystyle =\displaystyle 3x+8Reverse the division
\displaystyle 36\displaystyle =\displaystyle 3xReverse the addition
\displaystyle 12\displaystyle =\displaystyle xReverse the multiplication

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

Example 1

Solve the equation: -\dfrac{u}{4}+15=8

Worked Solution
Create a strategy

Apply the reverse operations in the reverse order.

Apply the idea

We can see from the equation that the expression on the left-hand side was built by applying the operations 'divide by -4', and 'add 15' to the pronumeral u.

So the order of reverse operations will be:

  1. Subtract 15

  2. Multiply by -4

\displaystyle -\dfrac{u}{4}+15\displaystyle =\displaystyle 8Write the equation
\displaystyle -\dfrac{u}{4}+15-15\displaystyle =\displaystyle 8-15Subtract 15 from both sides
\displaystyle -\dfrac{u}{4}\displaystyle =\displaystyle -7Evaluate
\displaystyle -\dfrac{u}{4}\times (-4)\displaystyle =\displaystyle -7 \times (-4)Multiply both sides by -4
\displaystyle u\displaystyle =\displaystyle 28Evaluate

Example 2

Solve the equation: \dfrac{8x+4}{5}=-4

Worked Solution
Create a strategy

Apply the reverse operations in the reverse order.

Apply the idea

We can see from the equation that the expression on the left-hand side was built by applying the operations 'multiply 8', 'add 4', and 'divide by 5' to the pronumeral x.

So the order of reverse operations will be:

  1. Multiply by 5

  2. Subtract 4

  3. Divide by 8

\displaystyle \dfrac{8x+4}{5}\displaystyle =\displaystyle -4Write the equation
\displaystyle \dfrac{8x+4}{5} \times 5\displaystyle =\displaystyle -4 \times 5Multiply both sides by 5
\displaystyle 8x+4\displaystyle =\displaystyle -20Evaluate
\displaystyle 8x+4-4\displaystyle =\displaystyle -20-4Subtract 4 from both sides
\displaystyle 8x\displaystyle =\displaystyle -24Evaluate
\displaystyle \dfrac{8x}{8}\displaystyle =\displaystyle \dfrac{-24}{8}Divide both sides by 8
\displaystyle x\displaystyle =\displaystyle -3Evaluate
Idea summary

To solve an equation 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

Outcomes

ACMNA194

Solve linear equations using algebraic and graphical techniques. Verify solutions by substitution

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