Linear Equations and linear Functions

Definition:

  

Linear Equations: A linear equation is defined as an algebraic equation in which every expression is either a stable (constant) or the product of a stable and a single variable. Linear equations can have one or more than one variables.  

  Linear functions:  A linear function is defined as a function whose chart (graph) consists of sectors of one straight line during its domain.

We can establish the linear function and it acquires value f (a) at ‘a’ and f (b) at ‘b’ by the following formula:
                                 
f(x)=f(a) (x-b)/(a-b) + f(b) (x-a)/(b-a)
   Because the first term is Zero (0) when x is ‘b’ and is f (a) when x is ‘a’, while the second term is Zero (0) when x is ‘a’ and is   f(b) when x is ‘b’.
   Finally we get,     y= mx+b


1. Example of Linear Equation:
Solve 3[2m - (7 - 3m)] = m - 21.           
Solution:
First we can solve the left hand side.
3[2m-(7-3m)]
Here first solving the inside term.
3(2m-7+3m) and then,
3(5m-7)
15m -21
Now compare with right hand side.
15m- 21= m-21
15m-m= 21-21
14m= 0
Therefore m=0
Answer: m=0

2. Example of Linear Function:
Solve 3(2m - 1) = 4(m + 5).
Solution:
3(2m - 1) = 4(m + 5)
6m-3=4m+20
6m-4m= 20+3
2m= 23
  m =23/2
  m =11.5
Answer: m=11.5