Introduction
- An algebraic expression is a mathematical expression that consists of variables, numbers and operations. The value of this expression can change Clarify the definitions and have students take notes on their graphic organizer.
Note:
- Algebraic Expression—contains at least one variable, one number and one operation. An example of an algebraic expression is n + 9.
- Variable—a letter that is used in place of a number. Sometimes, the variable will be given a value. This value will replace the variable in order to solve the equation. Other times, the variable is not assigned a value and the student is to solve the equation to determine the value of the variable.
- Constant—a number that stands by itself. The 9 in our previous vocabulary is an example of a constant.
- Coefficient—a number in front of and attached to a variable. For example, in the expression 5x + 3, the 5 is the coefficient.
- Term—each part of an expression that is separated by an operation. For instance, in our earlier example n + 9, the terms are n and 7.
Examples
Write each phrase as an algebraic expression.
Nine increased by a number r → 9 + r
Fourteen decreased by a number x → 1 4 − x
Six less than a number t → t− 6
The product of 5 and a number n → 5 × n or 5n
Thirty-two divided by a number x 32 ÷ x or 32/x
Example
An electrician charges sh 450 per hour and spends sh 200 a day on gasoline. Write an algebraic expression to represent his earnings for one day.
Solution
Let x represent the number of hours the electrician works in one day. The electrician's earnings can be represented by the following algebraic expression:
450x − 200
Simplification of Algebraic Expressions
Note:
Basic steps to follow when simplify an algebraic expression:
- Remove parentheses by multiplying factors.
- Use exponent rules to remove parentheses in terms with exponents.
- Combine like terms by adding coefficients.
- Combine the constants.
Like and Unlike Terms
- Like terms have the same variable /letters raised to the same power i.e. 3 b + 2b = 5b or a + 5a = 6a and they can be simplified further into 5b and 6a respectively(a2 and 3a2 are also like terms).While unlike terms have different variables i.e. 3b + 2 c or 4b + 2x and they cannot be simplified further.
Example
3a + 12b + 4a − 2b = 7a + 10b (collect the like terms )
2x − 5y + 3x − 7y + 3w = 5x − 12y + 3w
Example
Simplify: 2x – 6y – 4x + 5z – y
Solution
2x – 6y – 4x + 5z – y = 2x – 4x – 6y – y + 5z
= (2x – 4x) – (6y + y) + 5z
= − 2x – 7y + 5z
Note:
–6y – y = −( 6y + y)
Example
Simplify: 1/2 a − 1/3b + 1/4a
Solution
The L.C.M of 2, 3 and 4 is 1 2.
Therefore
1a − 1b + 1a = 6a − 4b + 3a
2 3 4 12
6a − 4b + 3a
12
=9a−4 b
12
Example
Simplify:
a+b − 2a − b
2 3
Solution
a+b − 2a − b = 3(a + b) − 2(2a − b)
2 3 6
3a + 3b − 4a + 2b
6
3a + 3b − 4a + 2b
6
= −a+5b
6
Example
- 5x² − 2x² = 3x²
- 4a²bc − 2a²bc = 2a²bc
- a²b – 2b3c + 3a²b +b3c = 4a²b – b3c
Note:
- Capital letter and small letters are not like terms.
Brackets
- Brackets serve the same purpose as they do in arithmetic.
Example
Remove the brackets and simplify:
- 3(a + b ) – 2(a – b)
- 1/3a + 3 (5a + b – c )
- 2b + 3{3 – 2 (a – 5)}
Solution
- 3(a + b ) – 2(a – b) = 3a + 3b – 2a + 2b
= 3a - 2a + 3b + 2b
= a + 5b - 1/3a + 3 (5a + b – c ) = 1/3a + 15 a + 3b – 3c
= 151/3 a + 3b − 3c - 2b + a { 3 – 2(a – 5 )} = 2b + a {3 – 2a + 1 0}
=2b + 3a – 2a2 + 1 0a
= 2b + 3a + 10a − 2a2
= 2b + 13a − 2a2
The process of remaining the brackets is called expansion while the reverse process of inserting the brackets is called factorization.
Example
Factorize the following:
- 3m + 3n
- ar3 + ar4 + ar5
- 4x2y + 20x4y2 − 36x3y
Solution
- 3(m + n) (the common term is 3 so we put it outside the bracket)
- ar3 + ar4 + ar5 (ar3 is common)
ar3(1 + r + r2) - 4x2y + 20x4y2 − 36x3y (4x2y is common)
= 4x2y( 1 + 5x2y − 9x)
Factorization by Grouping
- When the terms of an expression which do not have a common factor are taken pairwise, a common factor can be found. This method is known as factorization by grouping.
Example
Factorize:
- 3ab + 2b + 3ca + 2c
- ab + bx – a - x
Solution
- 3ab + 2b + 3ca + 2c = b(3a + 2) + c(3a +2 )
= (3a + 2) (b + c) - ab + bx – a – x = b (a + x) – 1 (a + x)
= (a + x) (b – 1)
Algebraic Fractions
- In algebra, fractions can be added and subtracted by finding the L.C.M of the denominators.
Examples
Express each of the following as a single fraction:
- x − 1 + x + 2 + x
2 4 5 - a + b − b − a
b a - 1 + 3 + 5
3(a+b) 8(a+b) 12a
Solution
- x − 1 + x + 2 + x = 10(x − 1) + 5 (x+2 ) + 4x
2 4 5 20
(10x − 10 + 5x + 10 + 4x)
20
= 19 x
20 - a + b − b − a = b(a+b) − a(b−a)
b a ab
= a² + b²
ab - 1 + 3 + 5 (find the L.C.M. of 3 , 8 and 12 which is 24)
3(a+b) 8(a+b) 12a
(find the L.C.M. of a and ( a+b) is a( a+b) )
(The L.C.M. of 3(a+b) , 8(a+b )and 12 is 24a(a+b ) )
= 1 + 3 + 5 =
3(a+b) 8(a +b) 12a
= 8a + 9a + 10a + 10b
24a (a+b)
=27a+10b
24a( a+b)
Simplification by Factorization
- Factorization is used to simplify expressions
Examples
Simplify
p²– 2pq + q²
2p²-3pq + q²
Solution
Numerator is solved first.
p² − Pq − pq + q²
p(p − q) − q(p + q)
(p−q)²
Then solve the denominator
2p² − 2pq − pq − q²
(2p − q) (p − q)
= (p−q)²
(2p − q)(p − q)
(p − q)²= (p − q)(p − q) hence it cancels with the denominator
Example
Simplify
16m² − 9n²
4m² − mn − 3n²
Solution
Num. (4m – 3n) (4m + 3n)
Den. 4m²– 4mn + 3mn – 3n²
= (4m + 3n) (m – n)
(4m – 3n)(4m + 3n)
(4m + 3n)(m – n)
4m – 3n
m – n
Example
Simplify the expression.
18xy – 18xr
9xr – 9xy
Solution
Numerator
18x(y – r)
Denominator
9x (r – y)
Therefore
18x(y − r)
9x(r − y)
= (y − r)
(r − y)
Example
Simplify
12x² + ax − 6a²
9x²− 4a²
Solution
(3x − 2a) (4x + 3a)
(3x+2a)(3x − 2a
= 4x + 3a
3x + 2a
Example
Simplify the expression completely.
ay − ax
bx − by
Solution
a(y − x) = a(y − x) = a = −a
b(x − y) −b(x − y) −b b
Note:
x − y = −(y − x)
Substitution
- This is the process of giving variables specific values in an expression
Example
Evaluate the expression
x² + y² if x =2 and y = 1
y + 2
Solution
x² + y² = 2² + 1² = 4 + 1
y + 2 1 + 2 3
= 5/3 = 12/3
Past KCSE Questions on the Topic
- Given that y = 2x – z express x in terms of y and z
x + 3z - Simplify the expression
x − 1 − 2x + 1
x 3x
Hence solve the equation
x − 1 − 2x + 1 = 2
x 3x 3 - Factorize a2 – b2
Hence find the exact value of 25572 - 25472 - Simplify
p²– 2pq + q²
P3 - pq²+ p²q – q3 - Given that y = 2x – z, express x in terms of y and z.
Four farmers took their goats to a market. Mohammed had two more goats as Koech had 3 times as many goats as Mohammed, whereas Odupoy had 1 0 goats less than both Mohammed and Koech.- Write a simplified algebraic expression with one variable, representing the total number of goats.
- Three butchers bought all the goats and shared them equally. If each butcher got 1 7 goats, how many did odupoy sell to the butchers?
- Write a simplified algebraic expression with one variable, representing the total number of goats.
- Solve the equation
1 = 5 − 7
4x 6x - Simplify
a + b
2(a+b) 2(a−b - Three years ago, Juma was three times as old as Ali. In two years time, the sum of their ages will be 62. Determine their ages
Download Algebraic Expressions - Mathematics Form 1 Notes.
Tap Here to Download for 50/-
Get on WhatsApp for 50/-
Why download?
- ✔ To read offline at any time.
- ✔ To Print at your convenience
- ✔ Share Easily with Friends / Students
