SUBJECT: Algebra
TOPIC: Expressions
LESSON: Linear Expressions
PREVIOUS KNOWLEDGE: Students can add, subtract, multiply or divide numbers. Students can add or subtract elements in a set.
TEACHING AIDS: A chart with some pre-written exercises to be used to introduce the lesson, 5 cakes, 6 pens and 8 books, white and colored chalk.
TEACHING METHOD: Demonstration, Questioning, Group Discussions, and Assignment.
OBJECTIVES: By the end of the lesson, students are expected to:
- define a linear expression and identify the nature of the terms therein.
- simplify linear expressions by combining like terms.
- factorize or expand linear expressions
GRADE LEVEL: 7 (Form 2)
DURATION: 45 Minutes
Introduction
Look at the chart on the board and give me answers to the following exercises;
i) 2+6=?
ii) 6-4=?
iii) 2cakes +3cakes=?
iv) 8books-5books=?
v) 2pens +4pens=?
vi) 2pens+3books=?
vii) 8books-3pens=?
If c represents a cake, b represents a book and p represents a pen then the above statements involving cake, book, and pen can be summarized as 2c+3c, 8b-5b, 2p+4p, 2p+3b, and 8b-3p. Statements like these are called linear expressions.
Presentation
Definition: A linear expression is a mathematical statement in which letters or variables are used for numbers. Example :9x, 2c+3c, x+3, 3x-4y, 3ab+2a+m. etc.
A linear expression is made up of terms. In 9x, the terms are 9 and x. In 2c+3c, the terms are 2,3 and c. Sometimes, a product is written as a product of letters and ordinary numeral (9x above). No equality sign exists in an expression
In 9x, 9 is called the numerical coefficient and the letter x is used for the number and is in no sense an abbreviation for any other thing. The 9 tells us the number of letters in the expression.
1) LIKE and UNLIKE TERMS in a linear expression
In the expression, 2x+3y+4x+7y, all the terms in x and y are like terms. That is 2x and 4x are like terms just like 3y and 7y. Like terms only differ in their coefficients and at times their signs. An expression like 2x+3y+4x+7y can be simplified by combining like terms. Also, the terms 2x and 3y are unlike terms; we cannot simplify any expression with unlike terms because the variables are different. The term 2x= x+x or 2*x..
2) SIMPLIFYING LINEAR EXPRESSIONS
To simplify a linear expression, add or subtract like terms. Where there are no like terms, the expression is left the same.
Example:
Simplify as far as possible the following:
i) 8x+2-3x+4.
ii) 2x+3y-4x+2y.
iii) 3x-xy+2xy+5x.
iv) 2ab-3a-ab+5b.
Solution
i) 8x+2-3x+4= 5x+6.
ii) 2x+3y-4x+2y=-2x+5y.
iii) 3x-xy+2xy+5x=8x+xy.
iv) 2ab-3a-ab+5b=ab-3a+5b.
3) SIMPLIFYING LINEAR EXPRESSIONS BY REMOVING BRACKETS
Remove the brackets by multiplying all terms in the bracket by the one outside.
Example:
Simplify each of the following expressions;
i) 3(x+6).
ii) 4(a+b).
iii) 3(x+2)+5(x-3).
iv) 4(a+2)+3(2a-1).
v) 2(6n+1)-3(n-5).
vi) 3(2x+3)-2(y-2).
Solution
i) 3(x+6)= 3x+18.
ii) 4(a+b)=4a+4b.
iii) 3(x+2)+5(x-3)=3x+6+5x-15
3(x+2)+5(x-) =8x+9.
4(a+2)+3(2a-1)=4a+8+6a-3
4(a+2)+3(2a-1)=10a+5.
v) 2(6n+1)-3(a-5)=12n+2-3n+15
2(6n+1)-3(a-5)=9n+17.
Vi) 3(2x+3)-2(y-2)=6x+9-2y+4
3(2x+3)-2(y-2)=6x+13-2y.
4) FACTORIZING LINEAR EXPRESSIONS
Factorizing a linear expression is to write it as a product of two or more terms. This is done by removing the common factor(s) using the distributive law of multiplication and insert brackets.
Example:
Factorize each of the following simplifying your result as far as possible.
i) 3a+3b+3c
ii) ax+ay
iii) 2xy-4x
iv) 12a+18ab
v) 6pn-8xn
Solution
i) 3a+3b+3c=3(a+b+c).
ii) ax+ay =a(x+y).
iii) 2xy-4x=2.x.y-2.2.x (remove common factors)
2xy-4x=2x(y-2).
iv) 12a+18ab=6.2.a+6.3.a.b (remove common factors)
12a+18ab=6a(2+3b).
v) 6pn-8xn= 2.3.p.n-2.4.x.n (remove common factors)
6pn-8xn= 2n(3p-4x).
5) EXPANDING LINEAR EXPRESSIONS
This is done by removing the brackets found in the expression then
Example:
Expand each of the following simplifying your result as far as possible.
i) 3(2a+3b)
ii) (a+x)(b+y)
iii) (x+y)(x+y)
iv) (a+b)(a-b)
Solution
i) 3(2a+3b)=6a+9b.
ii) (a+x)(b+y)=ab+ay+bx+xy.
iii) (x+y)(x+y)=x²+xy+xy+y²
(x+y)(x+y)=x²+2xy+y².
iv) (a+b)(a-b)=a²-ab+ab-b²
(a+b)(a-b)=a²-b².
Evaluation
a) Simplify the following:
i) 2x+3y+4x+5y.
ii) 2x+3-4y+5x.
iii) 2(y+2)+5(y-3).
iv) 2(6a+5)-5(a-2).
b) Factorize the following:
i) 2a+2b+2c.
ii) qx-qy.
iii) 12a+9ay.
iv) 6mn-8pm.
c) Expand
i) x(5y+p)
ii) (p+q)(s+r)
ii) (m+n)(m-n)
Summary
Any statement written as 5ax+2y is a linear expression 5, a,x,2, and y called the terms of the expression.5 and 2 are the coefficients while a,x and y are variables.
To simplify a linear expression, add or subtract like terms. Unlike terms are neither added nor subtracted.
To factorize, remove common factors and introduce brackets.
To expand, multiply all terms in the brackets by the one outside and combine like terms. The sign of the term outside should be considered.
Any expression that can be expanded can be factorized.
Conclusion
Exercises to be solved as assignments at home.
a) Simplify the following:
i) 3p+5q-2p-6q
ii) 4(3x-2)+5(x+4)
iii) 3(m+2)+4(m-3)
b) Factorize the following:
i) px+qx
ii) 2a+4b-6c
c) Expand each of the following:
i) 2(6x-4)
ii) -2(2n+3c-4)
iii) (m+a)(p+q)
iv) (r-s)(r+s)
Awah A.
Hi, I'm Awah! I have been teaching/tutoring since 1997 and specialize in Mathematics and Communications.
