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Australia
Year 7

8.05 Solving two-step equations

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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.

Use algebra to solve 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:

This image shows solving an equation using the reverse operations. Ask your teacher for more information.

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:

  1. Start with addition and subtraction

  2. Then multiplication and division

  3. 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:

  1. Add 8

  2. Divide by 7

  3. Subtract 6

Notice that we apply the reverse operation 'subtract 6' last since y+6 is contained within a pair of brackets.

This image shows solving a complicated equation using the reverse operations. Ask your teacher for more information.

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.

a

If we want to make n the subject of the equation, what is the first step we should take?

A
Add 15 to both sides to find the value of 5n.
B
Add 15 to only the left-hand side of the equation.
C
Divide only the left-hand side of the equation by 5.
D
Divide both sides by 5 to find the value of n-15.
Worked Solution
Create a strategy

Get the order of reverse operations.

Apply the idea

We can see on the left-hand side of the equation that 15 has been take away from n and the result has been multiplied by 5.

So we have the operations: the subtract 15 and multiply by 5.

So the order of reverse operations will be:

  1. Divide by 5

  2. Add 15

So the first step that we should take if we want to make n the subject of the equation is to divide both sides by 5 to find the value of n-15.

So option D is the correct answer.

b

What is the second step we should take to make n the subject?

A
Multiply only the left-hand side of the equation by 5.
B
Multiply both sides by 5 to find the value of 5(n-15).
C
Add 15 to both sides to find the value of n.
D
Add 15 to only the left-hand side of the equation.
Worked Solution
Apply the idea

In part (a) we have the order of reverse operations:

  1. Divide by 5

  2. Add 15

So the second step we should take to make n the subject is to add 15 to both sides to find the value of n.

So option C is the correct answer.

c

Apply the steps found in parts (a) and (b) to find the solution to the equation.

Worked Solution
Create a strategy

Apply the operations: divide by 5, then add 15 to both sides of the equation.

Apply the idea
\displaystyle 5(n-15)\displaystyle =\displaystyle 35Write the equation
\displaystyle \dfrac{5(n-15)}{5}\displaystyle =\displaystyle \dfrac{35}{5}Divide both sides by 5
\displaystyle n-15\displaystyle =\displaystyle 7Evaluate
\displaystyle n-15+15\displaystyle =\displaystyle 7+15Add 15 to both sides
\displaystyle n\displaystyle =\displaystyle 22Evaluate

Example 2

Solve the equation \dfrac{u+7}{2}=5.

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 'add 7' and 'divide by 2' to the pronumeral u.

So the order of reverse operations will be:

  1. Multiply by 2

  2. Subtract 7

\displaystyle \dfrac{u+7}{2}\displaystyle =\displaystyle 5Write the equation
\displaystyle \left(\dfrac{u+7}{2}\right)\times2\displaystyle =\displaystyle 5\times2Multiply both sides by 2
\displaystyle u+7\displaystyle =\displaystyle 10Evaluate
\displaystyle u+7-7\displaystyle =\displaystyle 10-7Subtract 7 from both sides
\displaystyle u\displaystyle =\displaystyle 3Evaluate

Example 3

Solve the equation 6=\dfrac{t}{5}-3.

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 right-hand side was built by applying the operations 'divide by 5' and 'subtract 3' to the pronumeral t.

So the order of reverse operations will be:

  1. Add 3

  2. Multiply by 5

\displaystyle 6\displaystyle =\displaystyle \dfrac{t}{5}-3Write the equation
\displaystyle 6+3\displaystyle =\displaystyle \dfrac{t}{5}-3+3Add 3 to both sides
\displaystyle 9\displaystyle =\displaystyle \dfrac{t}{5}Evaluate
\displaystyle 9\times5\displaystyle =\displaystyle \left(\dfrac{t}{5}\right)\times5Multiply both sides by 5
\displaystyle 45\displaystyle =\displaystyle tEvaluate
\displaystyle t\displaystyle =\displaystyle 45Swap sides
Idea summary

When applying reverse operations to isolate a pronumeral, we apply them according to the order of reverse operations:

  1. Addition and subtraction

  2. Multiplication and division

  3. 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.

Outcomes

ACMNA179

Solve simple linear equations

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