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5.01 Solving equations

Lesson

Introduction

When solving problems that involve missing or unknown values, we will often replace that value with a pronumeral in order to make an equation.

For example: Jill's age is equal to 5 more than twice the age of Jack. If Jill's age is 23, how old is Jack?

If we replace Jill's age with 23 and Jack's age with the pronumeral x, we can solve this question using the equation:23=5+2xTo find Jack's age, we can solve this equation to find the value of x.

Reverse operations

To solve equations using algebra, the most important rule to remember is that if we apply operations to one side of the equation, we must also apply it to the other.

When applying operations to equations, we always apply the same step to both sides of the equation. This way, both sides of the equation will be equal once we solve the equation.

Making sure to follow this rule, we can isolate the pronumeral in an equation by applying operations to both sides of the equation which reverse the operations applied to the pronumeral.

To do this, we need to know which operations are reversed by which.

OperationReverse operationExample
\text{Addition}\text{Subtraction}x+4-4=x
\text{Subtraction}\text{Addition}x-3+3=x
\text{Multiplication}\text{Division}y\times4\div4=y
\text{Division}\text{Multiplication}y\div 2\times 2=y

Examples

Example 1

Solve the equation: 24+p=59

Worked Solution
Create a strategy

The reverse of addition is subtraction, so we need to subtract from both sides of the equation to isolate p.

Apply the idea
\displaystyle 24+p\displaystyle =\displaystyle 59Write the equation
\displaystyle 24+p-24\displaystyle =\displaystyle 59-24Subtract 24 from both sides
\displaystyle p\displaystyle =\displaystyle 35Evaluate
Idea summary

When applying operations to equations, we always apply the same operation to both sides of the equation. This way, both sides of the equation will be equal once we solve the equation.

Order of reverse operations

If we return to the case of finding Jack's age, we can see that there are two operations that have been applied to the pronumeral in the equation:23=5+2x

As we can see, to get 23 we multiplied x by 2 and then added 5. In order to isolate 23 we need to reverse these operations in the correct order. Which operation should we try to reverse first?

If we try to reverse the multiplication first, we would divide both sides by 2 to get:\dfrac{23}{2}=\dfrac{5+2x}{2}

This only makes the pronumeral less isolated and is not what we want.

When reversing operations, we apply them in the reverse of the usual order of operations.

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

Knowing this, we can now solve the equation to find Jack's age.

\displaystyle 23\displaystyle =\displaystyle 5+2x
\displaystyle 18\displaystyle =\displaystyle 2xReverse the addition
\displaystyle 9\displaystyle =\displaystyle xReverse the multiplication

As such, we find that Jack's age is 9.

Examples

Example 2

Consider the equation 2(p+6)=-8

a

Which pair of operations will make p the subject of the equation?\begin{array}{ccccc} & \text{Step } 1 & & \text{Step } 2 & \\ 2(p+6) & \longrightarrow & p+6 & \longrightarrow & p \end{array}

A
Divide by 2, then subtract 6.
B
Divide by 6, then subtract 2.
C
Multiply by 2, then add 6.
D
Divide by 2, then add6.
Worked Solution
Apply the idea

Step 1 is the operation that can cancel out the multiplication of 2 which is divide by 2:\begin{array}{ccccc} & {\div2} & & \text{Step } 2 & \\ 2(p+6) & \longrightarrow & p+6 & \longrightarrow & p \end{array}

Step 2 is the operation that can cancel out the addition of 6 which is subtraction of 6:\begin{array}{ccccc} & {\div 2} & & {-6} & \\ 2(p+6) & \to & p+6 & \to & p \end{array}

So option A is the correct answer.

b

Apply these operations to the right-hand side of the equation as well.\begin{array}{ccccc} & \text{Divide by } 2 & & \text{Subtract } 6 & \\ -8 & \longrightarrow & ⬚ & \longrightarrow & ⬚ \end{array}

Worked Solution
Apply the idea

\begin{array}{ccccc} & \div2 & & -6 & \\ -8 & \longrightarrow & -4 & \longrightarrow & -10 \end{array}

c

Using your answer from part (b), what value of p will make the equation 2(p+6)=-8 true?

Worked Solution
Create a strategy

Combine the results in parts (a) and (b).

Apply the idea

In parts (a) and (b), we applied the same pair of operations to both the left-hand side and the right-hand side. We have:\begin{array}{ccccc} & {\div2} & & {-6} & \\ 2(p+6) & \longrightarrow & p+6 &\longrightarrow & p \\ \\ -8 & \longrightarrow & -4 & \longrightarrow & -10 \end{array}

If we apply the same operations to two equal expressions, then the results of those operations also equal.

So the value of p that will make the equation 2(p+6)=-8 true is p=-10.

Example 3

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 x+7-7\displaystyle =\displaystyle 10-7Subtract 7 from both sides
\displaystyle x\displaystyle =\displaystyle 3Evaluate
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

Move the pronumeral first

With the skills we now have, we can reverse as many operations as we need to isolate our pronumeral. But what if we want to solve equations like 4-x=17 or \dfrac{5}{x}=20?

In both of these cases we can see that the pronumeral is actually part of the operation being applied, so we can't isolate it so easily.

However, if we reverse the operation containing the pronumeral, we can move it so that the equation can be solved using the skills we just learned.

Examples

Example 4

Solve the equation \dfrac{-44}{p}=11.

Worked Solution
Create a strategy

Reverse the division of p first and then solve the equation.

Apply the idea
\displaystyle \dfrac{-44}{p}\displaystyle =\displaystyle 11Write the equation
\displaystyle \dfrac{-44}{p} \times p\displaystyle =\displaystyle 11 \times pMultiply both sides by p
\displaystyle -44\displaystyle =\displaystyle 11pSimplify
\displaystyle \dfrac{-44}{11}\displaystyle =\displaystyle \dfrac{11p}{11}Divide both sides by 11
\displaystyle p\displaystyle =\displaystyle -4Evaluate
Idea summary

If the pronumeral is actually part of the operation being applied, we can reverse the operation containing the pronumeral.

For example, in the following equation we are dividing by x: \dfrac{5}{x}=2 so we can use the reverse operation and multiply both sides by x: \begin{aligned} \dfrac{5}{x}\times x & =2 \times x \\ 5 &=2x \end{aligned}

Outcomes

VCMNA284

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

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