TYPES OF EQUATION
- Linear equation: linear equation is an equation involving equation involving variable with the highest power of one.
Linear equation is usually called by the number of variables involved in the equations. E.g. linear equation in one variable for example 2x + 3 = 0.
We also have linear equation in two variables example 2x + 4y = 5
And we also have linear equation in three variables e.g. 5X + 3y - z = 10 and so on
The variables are usually the letters involve in the equation - Quadratic equation: a quadratic equation is a single variable x is an equation of the second degree in x of the type ax2 + bx + c = 0 where a, b, c, are constant.
Example are: (i) x2 + 5x – 6 = 0
(ii) 2y2 + 3y + 4 =10 (iii) x2 + 2 = 0.
We have some special form of quadratic equation involving two variables.
Example: (1) 2x2 + xy – y = 5 (ii) x2 – 5xy + 3y2 =0.
Methods of solving equation: There is various method of solving equations, in most cases. It depends on the type of equation given.
- Linear equation: Method of solving linear equation is known as simultaneous linear equations which can either be solved by elimination or substitution methods. In solving simultaneous linear equation the number of variable involved should correspond with the number of equation given:
i.e. in case, the linear equation involve two variables than the number of equation given is two and so on.
Examples: solve the following simultaneous equations
(i) X – 3y = -2, 2x + 6y + 4 =0
(ii) X + y = 3, 2x – y = 3
(iii) X + 4y + 4z = 7, 3x + 2y + 2z = 6, 9x + 6y + 2z = 14
Solution: we are using elimination method for (i) and substitution method for (ii)
(i) X – 3y = -2……………….(1) X 2
2x + 6y + 4 = 0…………. (2)
Multiply equation (1) by 2.
2x - 6y = -4
+2x + 6y =-4
4x = -8
X= -2………….. (3)
Put equation (3) in (1)
-2 – 3y = -2
-3y = -2 +2
-3y = 2 +2
-3y = 0
Y = 0
x =-2 and y = 0
(ii)X + y = 3………….. (1)
2x – y = 3………….. (2)
From equation (1) we’ve
X = 3 – y……………………. (3)
Put (3) in (2)
2 (3-y) -y = 3
6 - 2y – y = 3
-3y = 3 – 6
-3y = -3
Y = 1…………(4)
Put (4) in three
X = 3-1
X =2.
X =2 and y= -1
(iii) X + 4y + 4z = 7…………………. (1)
3x + 2y + 2z = 6…………………….. (2)
9x + 6y + 2z = 14…………………… (3)
It is clear from the equation that it is easier to eliminate the variable z first
(3) – (2) gives 6x + 4y = 8 ……………………….(4)
(2) X (2) gives 6x + 4y + 4z = 12……………….(5)
(5) – (1) gives 5x = 5 x = 1
(5) – (4) gives 4z = 4 z =1
Using x = 1 in (4) 6 + 4y = 8
6y = 2 y = 1
Hence the solution are x = 1, y = 1, z = 1
Graphical method can be used in solving simultaneous equation especially in two variables. After plotting the graph of the two equations which gives a straight line their point of intersection is taking in the solution to the equation