year 9 mathematics - qed education · year 9 equations and inequalities mathematics theory booklet...

YEAR 9

MATHEMATICS EQUATIONS AND INEQUALITIES

THEORY BOOKLET

NAME: _____________

YEAR 9 EQUATIONS AND INEQUALITIES

MATHEMATICS THEORY BOOKLET

E: [email protected]

W: qededucation.com.au

T: 1300 044 066 p2

EQUIVALENT EQUATIONS

When solving an equation, we are really trying to find the value of the

pronumeral (i.e. the variable) that will make the equation true.

The correct value for the pronumeral will make both sides of the

equation balanced (equal), and thus, the solution satisfies the

equation.

To find the value of the pronumeral, we perform operations on the

equation to both sides. By performing the operation to both sides, the

equation does not change.

For example: if we add three to both sides…

Likewise we can perform addition (+), subtraction (-), multiplication

(x) or division ( ) operations so long as it is to both sides.

For example: if we divide both sides by 3…

Moreover solutions may also require multiple operations.

For example: if we subtract three from both sides… then if we divide both sides by 2… Remember, if we only performed out operations to one side, the

equation would become unbalanced and the sides would no longer be

equal. Performing operations one at a time to both sides

carefully changes the equation to the solution!

+3 +3

3 3

-3 -3

2 2





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Example: Solve the following equations.

i. ii.

iii. iv.





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Example: Using substitution, check if is the correct solution to

the equations.

i. ii.

Exercise: Solve the following equations.

i. ii. iii. iv.





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Exercise: Substitute the given solution to see if it satisfies the

equation. (i.e. check if it is the correct solution)

i. ii.

So far all our solutions to equations have been positive whole

numbers, however, solutions can also be fractions or negative.

For example: if we subtract six from both sides… then if we divide both sides by 10…

Moreover, our pronumerals don’t have to always be on the same side of the equation.

When this is the case we must first move all the pronumerals to one

side and constants (numbers without a pronumeral) to the other side

and collect like terms.

For example:

-6 -6

10 10





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i. ii. iii.


i. ii. iii.





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i. ii. iii. iv.





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1.1 EQUATIONS WITH GROUPING

SYMBOLS

If you remember how to expand grouping symbols, these equations

are no harder than the ones you have already seen

For example: ( ) ( ) if we expand the brackets on both sides… and collect like terms (subtract 2x from and add 18 to both sides) … then if we divide both sides by 4…


i. ( ) ii. ( ) ( ) iii. ( ) ( )

-2x, +18 -2x, +18

4 4





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i. ( ) ii. ( ) ( ) iii. ( ) ( )





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Example: Solve the following equations. Be careful with negative

signs and decimals!

i. ( ) ( ) ii. ( ) ( )





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Exercise: Solve the following equations. Be careful with negative

signs and decimals!

i. ( ) ( ) ii. ( ) ( ) iii. ( ) ( )





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2.1 EQUATIONS WITH FRACTIONS

When working with equations containing with fractions, we can

remove the fractions by multiplying by the denominator.

But make sure you multiply each term on both sides by the same

number.

Be especially when dealing with fractions that contain multiple terms.

For example consider the two cases below.

Case 1: Case 2:

if we multiply both sides by 2… if we multiply both sides by 2…

( )

and then solve normally… and then solve normally…

Example: Find the value of the pronumeral in each of the following

equations.

i.

ii.

x2 x2 x2 x2





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Exercise: Find the value of the pronumeral in each of the following

equations.

i. ii.

Example: Using substitution, show x=5 satisfies the equations.

i. ii.





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Exercise: Using substitution, check whether the given values for x

satisfy the equations.

i. ( )

ii. ( )

iii. ( )





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Example: Solve:

i.

ii. ( ) ( )

Exercise: Solve these equations with pronumerals on both sides.

i.

ii.

iii. ( )





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2.2 FURTHER EQUATIONS WITH

FRACTIONS

Previously we have only looked at equations with a single

denominator. However different terms in the equation can have

different denominators. In this case we must multiply by the

lowest common multiple of all the denominators. In other

words, we must multiply by a number that will cancel out every

denominator in the equation.

For example:

if multiply both sides by the lowest common multiple of 4 & 6 i.e. 12…

( ) ( )

and expand the brackets…

collect like terms…

and solve as normal…





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Example: Solve the following equations

i.

ii.





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Exercise: Solve the following equations. All solutions are integers.

Check your answer by substituting your solution into the equation.

i.

ii.

iii. iv.





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Example: Solve these harder equations.

i. ( ) ( )

ii. ( ) ( ) ( )





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Exercise: Solve these harder equations.

i.

ii.





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3.1 SOLVING PROBLEMS USING

EQUATIONS

Using equations to solve worded problems requires some careful

planning. First, you have to analyse the written problem, translate it

to an equation and then solve it.

Approach

Read the problem carefully and keep a look out for key words

Be aware of what you are trying to find, and what the given

information is (i.e. numbers)

Use the information to form an equation.

For example: “I multiple my age by 2 then add 6 giving me 30.”

So the equation is…

Example: For each of the following problems, form an equation and

then solve it.

i. I think of a number, triple it, add 4 and the result is 19. What is

the number?

ii. I think of a number. If I subtract 10 and then divide by 3 the

result is -1. Find the number.

Example: My mother was 24 years old when I was born and is now

three times as old me. How old am I?

+6 2 x a =30





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Sometimes, to solve questions we may need to recall equations we

already know. E.g. the equation for the perimeter of a rectangle, or

the area of a rectangle.

Exercise: Answer the following.

i. A rectangle is three times longer than it is wide. If it has a

perimeter of 104m, what are its dimensions?

ii. A rectangle is 12 cm longer than it is wide. Find its dimensions

if the perimeter is 120 cm.

iii. Peter and James have $24 between them. If Peter has three

times as much money as James, how much does he have?

iv. In a group of 16 friends there are 4 more brunettes then

blondes. Assuming there are no other hair colours, how many

brunettes are there in the group?

v. If a father is four times as old as his son at the moment. How

old is he now if he was 32 years old when his son was 2 years

old?





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Example: A car leaves Brisbane travels at 80km/hr towards

Melbourne. Another car leaves two and a half hours after, traveling

along the same path at 120km/hr. After how much time will the

second car catch up to the first? And how far from Brisbane would the

two cars be at this time?

Exercise: Roberto starts riding his bike from UNSW toward

Parramatta at noon, and Peter starts riding his bike from Parramatta

to UNSW at the same time. UNSW and Parramatta are 60km apart. If

Roberto rides at 25km/h but Peter only at 15km/h, find the time

when they meet.





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Example: Peter has won 82/90 tennis games between himself and

James. If Peter beats James in 90% of the games, he gets bragging

rights. If there are 10 games left in their tournament, how many

more games must Peter win to maintain bragging rights?

Exercise: Two rivers feed a a lake which holds 100, 000 litres of

water. The first river is a rapidly flowing river with a speed 20 times

as fast as the other. One very hot summer the lake empties out

(evaporation), but refils over a course of 5 rainy days. At what rate

did water flow into the lake?





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Exercise: Prize money is divided between first, second and third

place such that first place gets a half, second place gets a third and

third place gets the remaining (a sixth). If first place receives $500

more than second place, what was the total prize money?

3.2 INEQUATIONS

An inequation is a number sentence where the 'equals' sign

has been replaced by an inequality sign.

The inequality signs are:

> (greater than)

< (less than)

(greater than or equal to)

(less than or equal to)

Inequations, unlike equations, usually have more than one

solution. For instance:

the equation has one solution; the inequation has an infinite number of solutions.

The numbers are some solutions.

The full set of solutions is written as





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The solutions of inequations are often graphed on a number

line.

For example:

Note that because the solution set does not include , we have

not included on the graph (open circle). Had the solution set

been we would have coloured in the circle to signify that is a possible solution.

e.g.

Example: Solve the following inequations and graph the solution on

a number line.

i. ii. ( )

-2 6 2 0 4 8

-2 6 2 0 4 8





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Exercise: Solve the following inequations and graph the solution on a

number line.

i. ii. iii. iv.





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When multiplying or dividing by a negative number, the inequality

sign flips. i.e. an becomes a , and a becomes a , and vice

versa.

Example: Solve the following inequation and graph the solution on a

number line. Be careful with negatives!

i. ii. iii.

Exercise: Solve the following inequations and graph the solution on a

number line. Be careful with negatives!

i. ii. iii.





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Exercise: Solve and graph your solution.

i. ( ) ii. ( ) iii. ( )





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Example: Solve these harder inequations.

i. ii.

iii.

( )

iv. ( ) ( )





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Exercise: Solve these harder inequations.

i. ii.

iii.

iv.

( ) ( ) v.

( ) ( )





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3.3 QUADRATIC AND CUBIC EQUATIONS

So far we have only looked at linear equations, i.e. equations where

the variable is only to the power of 1. However this variable can have

a higher power.

Quadratic equations are equations where the variable is to the power

of 2.

These equations (and any others with an even power) typically have

two solutions.

There simplest form is: where is always positive (>0) and has two solutions: √ .

To find these two solutions we must square root the given value

(using a calculator), and then consider both the positive and negative

cases.

For example: if we square root both sides… √ √ but considering the negative case, we know that ( ) also

gives 25. So our solution is .

Cubic equations are equations with a variable to the power of 3.

These have only one possible answer.

There simplest form is: and has the solution √ .

To find the solution we must cube root the given value (using a

calculator).

For example: if we cube root both sides… √ √





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i. ii. iii.


i. ii. iii.





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i. ii.


i. ii.

iii.

s





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4.1 FORMULAE: EVALUATING THE

SUBJECT

Formulae are special types of equations. A formula represents a

relationship between physical quantities.

For instance, the formula represents the connection between

the area of a rectangle and its length and breadth.

A formula is different from an equation in that it will always

have more than one pronumeral. However, to find the value of a

pronumeral in a formula we must be told the values of every other

pronumeral in the formula. So there should always only be one

unknown.

Example: The potential energy ( ) of a mass is given by the formula where and are the mass and height above the ground of

the particle respectively. And is the gravitational acceleration. Find if , and

Example: Given the formula ( ) , find when and





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Exercise: The formula ( ) converts degrees Fahrenheit ( ) to degrees Celsius ( ). Find when:

i. ii. iii.

Exercise: The surface area ( ) of a sphere is given by the formula . Find the surface area correct to the nearest square

centimetre if the radius ( ) is 5 cm.





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Exercise: If , evaluate when and .

Exercise: If [ ( ) ], find when and

Exercise: Given that ( ) and ( ) find

and when and





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4.2 EQUATIONS ARISING FROM

SUBSTITUTION

Highlights:

• The isolated pronumeral that is on the left-hand side of an

equation or formulae is called the subject.

• We sometimes know the value of the subject and are asked to

find the value of one of the other pronumerals.

• To do this we should first rearrange the equation to make the

unknown the new subject then substitute values in.

Example: Answer the following problems.

i. The area of a triangle is given by , find when , and .

ii. The area of a trapezium is given by ( ). Find the value

of correct to 1 decimal place when . and .





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Exercise: Given , find:

i. when ii. when

Exercise: . Find:

i. when ii. when





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Example: For , find when and

Exercise: For the formula evaluate given that and





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Exercise: Given that ( ), find when and

Exercise: If , find when and





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4.3 SOLVING LITERAL EQUATIONS

Thus far, when provided a formula such as we have

substituted in given values and evaluated the subject ( ). Or if there

was another unknown, substituted in values, and rearranged the

equation for the unknown and then evaluated.

However, at times we are not given values for the pronumerals. In

this case we solve for the specified pronumeral as normal by

performing operations to both sides of the equation and rearranging

the equation to make it the subject.

Example: After expanding, solve each literal equation for . i. ( ) ii.





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Exercise: Solve each literal equation for the pronumeral shown in

the brackets.

i. [ ] ii. ( ) [ ] iii. ( ) [ ]





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Example: Make the subject of each formula.

i. √ ii. √

Exercise: Make the subject of each formula.

i. √ √ ii. √





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Example: Rewrite each equation making x the subject.

i. ii.

iii.





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Exercise: Rewrite each equation making the subject.

i. ii. iii. iv.

v.

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