# Stage 4 - algebraic techniques – solving equations

## Strategy

Students can:

• solve linear equations that may have non-integer solutions, using algebraic techniques that involve up to three steps in the solution process
• check solutions to equations by substituting

### Activities to support the strategy

To solve harder 2-step and 3-step equations, student first need to have confidence with the basic concepts of equations. These key concepts are the ideas of using inverse operations to ‘undo’ an algebraic expression and the need to keep an equation ‘in balance’ by ensuring that the same operation is always applied to both sides of the equals sign. When dealing with harder equations students need to also ensure that the correct order of operations is applied, especially when dealing with equations involving fractions. Students must also be reminded to take particular care with negative numbers. Lastly, the checking of solutions is often overlooked. This is a quick and easy step for students and is often an effective assessment strategy. A number of questions will need to be modelled to students and a significant amount of practice will be required to develop confidence.

### Activity 1

Students should initially revise their understanding of equations. In particular, the concepts of inverse operations and keeping the equation in balance need to be well-developed. Students should practice a number of 1-step and simple 2-step equations should be practised.

### Activity 2

The NSW syllabus provides a number of examples of 2-step and 3-step equations.

These examples should be modelled for students, with opportunities provided for students to solve a range of similar problems.

The solving fancier linear equations activity on Khan Academy provides a series of online tutorials and a number of questions involving the more complicated aspects of solving equations such as dealing with variables on both sides of the equation and equations involving fractions.

### Activity 3

The checking of solutions is often overlooked by students. This is a quick process and should be encouraged by students to test their answers. Checking of solutions involves simply substituting their answers back into the original question. It can also be an effective test strategy for students completing multiple choice questions. Often checking each answer in the original question provides a quicker result.

An example of one such multiple choice question is shown below.

Which of the following is the correct solution to 3 (x−1) 2 =15

(A)  x=12

(B)  x=11

(C)  x=10

(D)  x=9

Solution:
Instead of solving the equation, simply check each possible answer by substitution

A B

Checking (A) x=12

3(12−1)2 =3×112 =332 ≠15

Not (A)

Checking (B) x=11

3(11−1)2 =3×102 =302 =15

(B) Is the correct solution

## References

### Australian curriculum

ACMNA194: Solve linear equations using algebraic techniques and verify solutions by substitution

### NSW syllabus

MA4-10NA: Uses algebraic techniques to solve simple linear and quadratic equations