 Middle Years

# 6.01 Linear equations

Lesson

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$5 more than twice the age of Jack. If Jill's age is $23$23, how old is Jack?

If we replace Jill's age with $23$23 and Jack's age with the pronumeral $x$x, we can solve this question using the equation:

$23=5+2x$23=5+2x

To find Jack's age, we can solve this equation to find the value of $x$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.

Applying operations to equations

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.

Operation Reverse operation Example

Subtraction

Subtraction

$x+4-4=x$x+44=x

Multiplication

Division

Division

Multiplication

$y\times4\div4=y$y×4÷​4=y

#### Worked Examples

##### Example 1

Solve the equation: $x+\frac{2}{5}=\frac{17}{20}$x+25=1720

Think: To isolate $x$x, we want to reverse the addition of $\frac{2}{5}$25.

Do: Since subtraction reverses addition, we can isolate $x$x by subtracting $\frac{2}{5}$25 from both sides of the equation. This gives us:

$x=\frac{17}{20}-\frac{2}{5}$x=172025

Noting that $\frac{2}{5}$25 is equivalent to $\frac{8}{20}$820, we can perform the subtraction to find that:

$x=\frac{9}{20}$x=920

##### Example 2

Solve the equation: $\frac{x}{3.5}=4$x3.5=4

Think: To isolate $x$x, we want to reverse the division by $3.5$3.5.

Do: Since multiplication reverses division, we can isolate $x$x by multiplying both sides of the equation by $3.5$3.5. This gives us:

$x=4\times3.5$x=4×3.5

Performing the multiplication gives us:

$x=14$x=14

Reflect: In both cases, the original operation applied to the pronumeral was cancelled out by the reverse operation which we applied to both sides of the equation, leaving just $x$x on the left hand side.

### The 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$23=5+2x

As we can see, to get $23$23 we multiplied $x$x by $2$2 and then added $5$5. In order to isolate $x$x 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$2 to get:

$\frac{23}{2}=\frac{5+2x}{2}$232=5+2x2

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.

Order of reverse operations

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

2. Multiplication and division
3. Expressions inside brackets

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

 $23$23 $=$= $5+2x$5+2x $18$18 $=$= $2x$2x Reverse the addition $9$9 $=$= $x$x Reverse the multiplication

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

#### Practice questions

##### Question 1

Consider the equation $2\left(p+6\right)=-8$2(p+6)=8.

1. Which pair of operations will make $p$p the subject of the equation?

 Step $1$1 Step $2$2 $2\left(p+6\right)$2(p+6) $p+6$p+6 $p$p

Divide by $2$2, then subtract $6$6

A

Divide by $6$6, then subtract $2$2

B

Multiply by $2$2, then add $6$6

C

Divide by $2$2, then add $6$6

D

Divide by $2$2, then subtract $6$6

A

Divide by $6$6, then subtract $2$2

B

Multiply by $2$2, then add $6$6

C

Divide by $2$2, then add $6$6

D
2. Apply these operations to the right-hand side of the equation as well.

 Step $1$1 Step $2$2 $-8$−8 $\editable{}$ $\editable{}$
3. Using your answer from part (b), what value of $p$p will make the equation $2\left(p+6\right)=-8$2(p+6)=8 true?

##### Question 2

Solve the equation $\frac{x+15}{2}=5$x+152=5.

1. Enter each line of working as an equation.

### Moving 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$4x=17 or $\frac{5}{x}=20$5x=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.

#### Worked Example

Solve the equation: $4-x=17$4x=17

Think: We want to first reverse the subtraction of $x$x first and then solve the equation as per usual.

Do: We can reverse the subtraction by adding $x$x to both sides of the equation. This gives us:

$4=17+x$4=17+x

Since the order of numbers in addition doesn't matter, we can use the equivalent equation:

$4=x+17$4=x+17

Which we can solve by reversing the addition to get:

$-13=x$13=x

Reflect: Another way to solve this equation would be to reverse the addition of $4$4, then reverse the negative sign on the pronumeral by multiplying both sides of the equation by $-1$1. Doing this would follow the steps:

 $4-x$4−x $=$= $17$17 $-x$−x $=$= $13$13 Reverse the addition $x$x $=$= $-13$−13 Reverse the negative

We can solve equations where the pronumeral is the denominator using a similar method. To solve these equations we can multiply both sides of the equation by the pronumeral and then perform the usual reverse operations to isolate it.

#### Practice question

##### Question 3

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

1. Enter each line of working as an equation.