Working with Unknowns

Slideshow 8, MathematicsMr. Sasaki, Room 307

Objectives• Understand algebraic notation and

how to avoid using and symbols• Simplify simple algebraic

expressions• Express simple worded problems

with algebra

Purpose of AlgebraWhat is algebra?Algebra is a system where numbers are represented by symbols.What kind of

symbols?Usually letters (Latin characters) like…𝑥 𝑦 𝑎 𝑏 𝑐 𝑚 𝑛 𝑝 𝑞

Of course others are used too in specific cases. Also some Greek characters are used.Why is algebra important?Algebra makes solving problems easier. We can express, compare, summarise and make decisions with algebra. We can do lots more too!

Unknowns and NotationIn algebra, we call symbols unknowns when they represent one value (one number).At times, symbols refer to a group of numbers or all numbers. They are then called .

variables

We can then usually find out what the number is.

In algebra, appears often. You need a style to write it, make sure it doesn’t look like (the multiplication symbol)!Note: In fact, in algebra, we don’t usually write at all. (We don’t write either.)

Algebraic NotationIn Chapter 2, we will not solve many equations, just simplify expressions.Let’s use to help us represent the cost of some items (they cost \).

3 𝑥3 bananas Each costs \ and there is 3 of them.

7 𝑥7 apples Each costs \ and there is 7 of them.

Note: We don’t need to write or . The symbols disappear. and .

Algebraic NotationLet’s consider some more items that cost \ each.

𝑥4

A quarter of a cake

A whole cake costs \ so we consider .

Note: We shouldn’t write symbols either. In algebra, or is more acceptable. 2𝑥

3Two-thirds of a pizza

Writing is also fine. Don’t write

Note: as a decimal should be written as .

0. 6̇

Algebraic NotationA pizza costs \ and we need to pay an extra \ for delivery. How can we express this?𝑥+300 and symbols are used

frequently in algebra.Four people buy 2 bananas and 3 coconuts each. 4(2𝑥+3 𝑦 )Here, we assumed bananas cost \ and coconuts cost \.Note: An assumption is something we think is true without being told it is.

Brackets are used for grouping.

4× (2 𝑥+3 𝑦 )=¿

Answers - Easy3 𝑥 4 𝑦𝑥2 or

3 𝑥 3𝑥

6 𝑥 3 𝑥2 𝑦3

𝑥𝑦 𝑥2 𝑥2𝑦3 𝑥2 2 𝑥 3𝑥𝑦

2𝑥𝑦

2𝑥𝑦

3 (𝑥+𝑦)𝑥+2 𝑦2

2𝑥+3 𝑦2

2(𝑥+𝑦 )(𝑥+𝑦 )22 𝑥 (𝑥+𝑦 )2

Answers - Hard−𝑥 −𝑥−3 𝑥 0

1 0

−2+𝑥 −5 𝑥− 𝑦

2𝑥2𝑥 𝑦2

3𝑥2 𝑦

1𝑥2

4𝑥2

𝑥𝑦2𝑧

−8𝑥 9 𝑥2 𝑥𝑦𝑥

3𝑥𝑦

4 𝑥29

Worded ProblemsAlgebra is useful for modelling. A sentence looks messy compared with an algebraic expression.ExampleShunya buys 4 cars and 2 buses. Represent the cost of this algebraically.

4 𝑥+2 𝑦We let the cost of a car be \ and a bus be \. What did we assume?We assumed all of the cars cost the same and busses cost the same.

Worded ProblemsAlgebra can represent anything, not just cost. It could be any quantity or measurement.ExampleMinori makes 2 Youtube videos minutes long and mixes 3 songs minutes long. How much footage does he have in total?

2 𝑥+3 𝑦Note: A problem like this could be useful to test whether Minori’s footage fits on a CD or not.

≤90

The inequality above should be true for a 90min CD.

Answers3 𝑥+𝑦7 𝑥+𝑦2 𝑥− 𝑦3 𝑥+2 𝑦+𝑧

or or 3 𝑥+2 𝑦4 𝑥=5000

2 𝑥+ 𝑦=10 𝑧

2000=𝑥+1250

3 (2 𝑥+5 𝑦+2 𝑧 )=3500

or