Simultaneous Equations

Equations
Any expression F(X) when equated to an expression or a constant is called an equation.
When the highest power of X is one it is called a simultaneous equation.

What are Simultaneous Equations
A set of equations having many variables, is called simultaneous equations. To solve a simultaneous equation of two variables we need minimum two equations, which are different from each other.

Simultaneous Equation

Ex: 2x+3y = 5 and 4x +3y = 10.

Solving Simultaneous Equations:

Substitution method
Algorithm:
Step 1: Express one of the variables in terms of the other.
Step 2: Substitute the value of the variable in the second equation to obtain an equation in terms of one single variable.
Step 3: Solve for the variable.
Step 4: Substitute the value of the variable in the expression to get the solution of the other variable.

Solved Example:
Solve 2x +3y =5; 5x +2y=7
Step 1: 2x= 5-3y; x= (5-3y)/2
Step 2: Substituting the value of x in the second equation we get
 5((5-3y)/2) + 2y=7 Solving for y we get
 (25-15y+4y)/2=7
 (25-11y)/2=7
 25-11y=14
 25-14=11y
 11=11y
 y=1
Step 3: Substituting the value of x in step 1 we get x=(5-3(1))/2 =1
Hence the solution set is X=1, y=1 or {1, 1}

Method 2:
Elimination Method:
Algorithm:
Step 1: Make the coefficients of one of the variables same.
Step 2: If the signs are same, subtract and if the signs are different add them to obtain a linear equation of one variable and solve for it.
Step 3: Substitute the obtained value in any one of the equations to solve for the other variable.
Solved Example:
4x+ 3y = 11; 3x +2y=8
4x+3y=11  --------> multiply 2
3x+2y=8  --------> multiply 3
8x + 6y =22
9x + 6y =24 (as the signs are same subt)
(-)   (-)     (-)
-x = -2 or x=2
Substituting the value of x in the first equation we get
4(2) +3y =11;
8+3y=11;
3y= 11-8;
3y=3; y=1.
The solution set is x= 2 and y = 1.
Cross Multiplication Method
Algorithm:
Step 1: Rearrange the equations as shown.
If the equations are as follows

Step 2: Cross-multiply and subtract starting from top end corner of the denominator and write as follows

Step 3: 
Equate the x term with constant term and solve for x. Similarly equate y term with constant term and solve for y.
Solved Example:
2x +3y = 7
6x -5y =11

Cross-multiplying and subtracting we get

Product Rule for Derivatives

Product Rule for Derivatives:

The Product Rule helps us to differentiate a function when the function is a product of two distinct functions.

 Product Rule is necessary whenever we need to differentiate a function which is a product of two sub-functions that are easier to differentiate.

Let us consider some examples for a better understanding of the Product Rule for Derivatives

Associative Property

Definition of associative property:

Associative property pertains to number, sets and many other mathematical components. The word associative is derived from the word ‘association’, which means being related to. In math, we define associative property for various operations on numbers or sets etc.

Associative Property

What is associative propery?

All types of numbers, whether natural, whole, rational, irrational and real numbers have associative properties. There are mainly two types of associative property for numbers. They are:

1. Associative property of addition:
Addition associative property definition is as follows: Consider three numbers a, b and c. then, (a+b) + c = a + (b+c). Meaning we associate a with b first then associate the sum with c or we associate a with sum of b and c, either ways, the result would be the same.
For example: (1/2 + 1/3) + ¼ = ½ + (1/3 + ¼) = 13/12

2. Associative property of multiplication:
Numbers obey associative property in multiplication as well. Consider three numbers, p,q and r, then (p x q) x r = p x (q x r). That means whether we take product of p and q and then multiply with r or we multiply p with product of q and r, either ways we would get the same answer.

For example: (1/2 x 1/3) x ¼ = ½ x (1/3 x ¼) = 1/24
Associative property with respect to sets:

Consider three non disjoint sets, A, B and C. Then both the operations of union and intersection on these sets is associative. That means, A U (B U C) = (A U B) U C and
A ∩ (B ∩ C) = (A ∩ B) ∩ C.

Example: Consider the following sets: A = {1,2,3,4,5}, B = {2,4,6,8,10}, C = {3,4,5,6,7}
Then, A U (B U C) = {1,2,3,4,5} U ({2,4,6,8,10} U {3,4,5,6,7}) = {1,2,3,4,5,6,7,8,10}
And (A U B) U C = ({1,2,3,4,5} U {2,4,6,8,10}) U {3,4,5,6,7} = {1,2,3,4,5,6,7,8,10}
So we see that both are equal. Similarly .
A ∩ (B ∩ C) = {1,2,3,4,5} ∩  ({2,4,6,8,10} ∩  {3,4,5,6,7}) = {4,6} and
(A ∩ B) ∩ C = ({1,2,3,4,5} ∩  {2,4,6,8,10}) ∩  {3,4,5,6,7} = {4,6}