This document provides an overview of algebraic fractions including simplifying, operations, solving equations and systems of equations, and undefined values. Objectives covered include simplifying algebraic fractions, performing operations such as addition and subtraction, solving equations and simultaneous equations, making substitutions, and representing algebraic fractions graphically. Examples are provided for each topic to demonstrate the concepts and steps involved.

1. Algebraic
Fractions

2. Objectives
Revision On:
• Simplify of Algebraic Fraction
• Perform Operations on Algebraic Fraction
• Solve Equations Involving Algebraic Fraction
• Make Substitution in Algebraic Fraction
• Solve Simultaneous Equations Involving Algebraic Fraction
• Undefined value of an Algebraic Fraction
• Represent Algebraic Fractions Graphically.

3. Simplification of Algebraic Fractions
Like in common fractions, algebraic fractions can be reduced to their lowest form, by
dividing numerators and denominators by their common factor(s).
Example 1
Reduce
𝑏2−16
𝑏2−4𝑏
to its lowest form.
Solution
𝑏2 − 16=
𝑏2 − 42 = (𝐛 + 𝟒)(𝐛 − 4)
𝑏2
− 4𝑏= b(b – 4)
(𝐛+𝟒)(𝐛 − 4)
b(b – 4)
=

4. • Example 2
Simplify
𝑥2−8𝑥+16
𝑥2−7𝑥+12
Solution
Factorize numerator 𝑥2 − 8𝑥 + 16=
(x – 4)(x – 4)
Factorize denominator 𝑥2
− 7𝑥 + 12 =
(x - 4)(x – 3)
𝑥2−8𝑥+16
𝑥2−7𝑥+12
=
(x – 4)(x – 4)
(x − 4)(x – 3)
=

5. Operations on Algebraic Fractions
Example 3
Simplify
3
𝑚+2𝑛
+
2
𝑚 −3𝑛
Solution
LCM of (m+2n) and (m – 3n) is
(m+2n)(m – 3n)
=
3
𝑚+2𝑛
+
2
𝑚 −3𝑛
=
3 𝑚−3𝑛 +2(𝑚+2𝑛)
(m+2n)(m – 3n)
=
𝟓 𝒎−𝒏
(m+2n)(m – 3n)

6. Equations Involving Algebraic Fractions
Example 4
Solve the equation
1
2𝑘+1
+
1
2𝑘−3
−
1
𝑘−1
= 0
Solution
Multiply each by LCM of the denominators
(2k+1)(2k-3)(k-1)
1
2𝑘+1
(2k+1)(2k-3)(k-1)+
1
2𝑘−3
(2k+1)(2k−3)(k−1)+ −
1
𝑘−1
(2k+1)(2k-3)(k-1)=0
= (2k-3)(k-1) + (2k+1)(k-1) − (2k+1)(2k-3)
K = 2 ½

7. Solve Simultaneous Equations Involving
Algebraic Fraction
5𝑥
8
−
𝑦
2
=
1
4
2𝑥
3
−
3𝑦
5
=
2
15
STEPS:
1. Find the LCM of the
denominators of each equation.
2. Multiply through by the LCM to
clear the fractions.
3. Solve both resulting equations
simultaneously

8. • Example 5
Solve the following pair of equations.
5𝑥
8
−
𝑦
2
=
1
4
2𝑥
3
−
3𝑦
5
=
2
15
The LCM is 8. Multiply through by 8.
5𝑥 − 4𝑦 = 2
The LCM is 15. Multiply through by 15.
10𝑥 − 9𝑦 = 2

9. After clearing the fractions, solve simultaneously
𝑥 = 2 , 𝑦 = 2
5𝑥 − 4𝑦 = 2 __________ (1)
10𝑥 − 9𝑦 = 2 ___________ (2)

10. • Example 6
Solve
1+2𝑦
𝑥
= 5 𝑎𝑛𝑑
3+4𝑦
2𝑥
= 3
1
2
LCM of denominator of first equation is
LCM of denominator of second equation is
Multiply both side of first equation by x
Multiply both side of second equation by 2x
1 + 2y = 5x
3 + 4y = 7x
x
2x

11. 1 + 2y = 5x ________ (1)
3 + 4y = 7x ________ (2)
After clearing the fractions, solve simultaneously
X = −
1
3
𝑦 = −1
1
3

12. Classwork
Solve the following simultaneous equations
1. 𝑥+𝑦
4
=
1
2
𝑎𝑛𝑑
𝑥−3𝑦
3
=2
2.
𝑥
2
+
𝑦
4
=
1
2
𝑎𝑛𝑑
𝑥
3
−
𝑦
5
=
1
9
ans: x=3, y=-1
ans: x=
23
33
, y=
20
33

13. Read Up
• Make Substitution in Algebraic Fraction
• Undefined value of an Algebraic Fraction
Reference: MAN Mathematics SS 2

14. Undefined Value of Algebraic Fraction
• For algebraic fraction to be undefined, the following
conditions must apply:
1. The numerator and denominator must have no common
factor(s)
2. The denominator must be equal to zero.

15. • Example 7
Find the value of x for which
4𝑥
5𝑥+1
is undefined.
Solution
Since the numerator and denominator doesn’t have any common factor, equate
the denominator to zero.
5𝑥 + 1 = 0
Solve for x,
X= - 1/5

16. Example 8
Find the value of x for which
𝑥2−4
𝑥−2
is undefined.
Solution
Since there is a common factor, factorize first.
(𝑥 − 2)(𝑥 + 2)
𝑥 − 2
The equate the denominator to zero and solve for x:
Since there is no x in the denominator, then x has no value.

17. Classwork
• Find the value of x for which the following algebraic fractions are undefined.
1. 3𝑥+7
10−2𝑥
2. 2𝑥−15
6𝑥
3. 𝑥+2
𝑥2−5𝑥 −14

18. Correction
1. 3𝑥+7
10−2𝑥
10 – 2x = 0
2x = 10
X = 5
2.
2𝑥−15
6𝑥
6x = 0
X = 0
3.
𝑥+2
𝑥2−5𝑥 −14
Factorizing 𝑥2 − 5𝑥 − 14
(x+2) (x -7)
𝑥+2
(x+2) (x −7)
=
1
𝑥 −7
X -7 = 0
X =7

19. Graphical Interpretation of Undefined Value

21. References
• https://www.youtube.com/watch?v=EPM5XEXKa_Y
• https://www.youtube.com/watch?v=2N62v_63SBo
• https://www.youtube.com/watch?v=vA-55wZtLeE
Email: hammedalao@gmaill.com