Topic 12: Algebra

By the end of this topic, you will be able to:

• use letters to represent numbers
• write statements in algebraic form simplify algebraic expressions
• evaluate algebraic expressions by substituting numerical values
• manipulate simple algebraic equations in one variable and solve them

Keywords

• algebraic
• coefficient
• evaluation
• expression
• letter
• like term
• number
• simplification
• statement
• substitution

Introduction

As we apply mathematics in our daily life activities, sometimes some of the parameters may not be known to us. For instance if you spend UGX 500 daily at break time, how much do you spend in 2 days? What if you are not sure about the number of days?

Is there any [removed]formula) you can use to find the amount for number of days? Or if you have a party and you invite 10 friends planning will be easy, but what if you don’t know the number of guests, how will you plan? Is there any formula you can use to establish how many crates of soda, how much beef, rice, etc, that you should buy?

Such challenges call for use of symbols, letters in most cases, to represent the unknown. The resulting mathematical expression is referred to as algebraic expression. In this topic therefore, you will learn to form and use simple algebra expressions and use them to solve everyday life problems.

Using Letters to Represent Numbers

Activity 12.1 Using letters to represent numbers (work in groups)

What you need: 10 small boxes, pens with different colours, a pen and a note book.

What to do:

1. Put the same number of pens in each box; 2 boxes for red pens; 3 boxes for black pens and 5 boxes for blue pens.

2. Considering the number of pens in each box as an unknown represented by any letter of your choice what is the total number of pens?

3. Present your answer to the class.

Example

Imagine you have 4 cups with pencils inside them and 5 pencil on the table. Assuming all the cups have an equal number of pencils inside, how many pencils are there altogether? Let the number of pencils in each cup be

x. x+x+x+x+5 (4x + 5) pencils.

Exercise 12.1

1. Three students wanted to keep their pens together in one box. John puts twice the number of pens as Kate. Jacob put three more pens than Kate. How many pens were put in the box?

2. Kawooya bought 19 books. Musana bought 3 books, Isabirye bought 5 books more than Mulwana. How many books did they buy altogether?

Writing Statements in Algebraic Form

Activity 12.2 Writing statements in algebraic form (work in groups)

What you need: a manila paper, markers, pens, notebook.

What to do:

1. Draw a number machine on a manila paper.

2. Put different mathematical operations of your choice in the number machine.

3. Use letters of your choice to represent unknown numbers, write down the algebraic expression generated by your number machine.

4. Present your work to the class.

5. Hang the chart in a corner inside the classroom.

6. Compare your work with other groups’ work.

Example

Muwonge has x books in his bag and 5 books his hand. His friend Auma has 4x books in her hand and 8 books in her bag. How many books do the two friends have altogether?

Solution

Muwonge = (x+5) books

Auma = (4x+8) books

Muwonge and Auma

= (x+5)+(4x+8)

= 5x + 13

Exercise 12.2

1. Suppose the unknown is q, write the algebraic expression for the following statements.

a) Sum of unknown number and 18.
b) The number that exceeds the unknown by 5.

c) The difference between 20 and the unknown number.

d) A third of the unknown number plus 8.

e) Four times the unknown number less twice the number.

2. By letting the unknown to be m, write the expression for

a) 3 more than m

b) Six less m

c) The sum of -6 and m

d) Ten times m.

e) Number which is twice m.

3. Use the given number machines to write the equivalent algebraic expressions.

Simplifying Algebraic Expressions

To simplify algebraic expressions, collect like terms together before applying the relevant mathematical operations.

Example 1

Simplify: 4y-6x+5y + 8x

Collect the like terms, add and finally subtract.

4y+5y+8x-6x

9y+ 2x

Example 2

Simplify: 2(3x+2y) – 3(x – y)

6x+4y-3x + 3y

6x-3x+4y+ 3y 3x + 7y

Exercise 12.3

1. Simplify the following algebraic expressions.

a) 8x + 7y – 5x + 2y.

c) 4xy + 3xy + 7xy + 14xy.

e) 2a + 3 + 5a + 3a + 3.

b) 14p-16q + 5p + 6q.

d) (c+3)+c+ 2.

f) 5b+2p+ 3c + 4c.

2. Simplify the following algebraic expressions.

a) a xbx c

c) 4(2x + 3y) + 2(x + y)

e) Add 3(4x+3y) to 2(5x + 3y)

g) 18x ÷ 3xy b) 3(2k – 4q)

d) 3(4p-2q)-3(p + q)

f) 129 ÷ 3 h) -6m -2m

Evaluating Algebraic Expressions by Substituting Numerical Values

In an algebraic expression such as 2a + 3b, a and b are not known, when you are given their values, then a and b will be replaced by their respective values, this process is called substitution and calculating to get the final value of the expression is called evaluation.

Activity 12.3 Evaluating algebraic expression by substituting numerical values (work in groups)

What you need: pens, notebook.

What to do:

1. Let each member in your group come up with two or more algebraic expressions.

2. Give different values for each unknown in the corresponding algebraic expression formed.

3. Substitute and calculate to get the value of the expression.

4. Present your work to the whole class.

Example

3 people won a cash prize of UGX P. What amount will each one get if:
a) They share it equally.

b) One of them takes 3 times as much as the other 2?

c) If the prize amount was UGX.450000, how much did each get in a) and b) above?

Solution

a) UGX p/3 b) 2 people = k 1 person = 3k

Manipulating and Solving Simple Algebraic Equations

You manipulated some equations of one variable in your primary mathematics. These equations involve subtraction, addition, division and multiplication.

Example

Forming and Solving Equations

Activity 12.4 Forming and solving equations

What you need: a notebook, pen.

What to do:

1. Think of a number add 4, triple the result, the answer is 36, find the number.

2. Form many other equations and solve them.

3. Present your findings to the whole class.

Example 1

Ronna had a certain number of mangoes, she got more 6 mangoes from Amina, then Mutagubya took 8 mangoes from her. How many mangoes did Ronna have at first if she was left with 16 mangoes altogether?

Let the original number of mangoes Ronna had be y

y+6-8=16

y -2 = 16

y-2+2=16 + 2

y = 18

.. Ronna had 18 mangoes altogether

Example 2

Excercise 12.6

1. Think of a number, subtract 4 from it, triple the result, then add 15, the answer is 12. What is the number?

2. Musana had a certain number of pencils, he bought 3 more pencils, doubled the result and gave 9 pencils to Alister and remained with 13 pencils altogether. al Construct an equation to represent Musana’s number of pencils. b) How many pencils did Musana have at first?

3. Okyamolinga had a certain number of oranges, he ate 7 oranges in the morning, he was given more oranges equal to those he had remained with. The number of oranges became 16.

a) Form an equation representing the above information.

b) How many oranges did he have at first?