Concept of Linear Equation for SSC and Competitive exams

Linear Equation means finding the value of Y for the given value of X. If the equation is in the form of Y = MX + B, with Y & X are the variables, and M & B are the rational numbers.

General Form:

In general form, y = mx +b where y and x can not be zero and y>0. The graph of the equation is a straight line, and every straight line can be represented by an equation in the above form.

Slope-Intercept form:

y = mx + b  or ax + by + c = 0
where, m is the slope or gradient and b is the Y-intercept constant.

for example: y = 2x +1, then m = 2 & b = 1. or 
3y = 2x - 3, then y = (2/3)x -3/2 => m = 2/3 and b = -3/2

Conditions for Solvability:

y = mx + b  or ax + by + c = 0
the system of equation is ax1 + by1 + c1 = 0 and ax2 + by2 + c2 = 0

(i) a unique solution, if a1/a2 ≠ b1/b2
(ii) an infinite number of solutions, if a1/a2 = b1/b2 = c1/c2
(iii) no solution, if a1/a2 = b1/b2 ≠ c1/c2 

Consistent System:- A system consisting of two simultaneous linear equations is said to be consistent, if it has one solution.
Inconsistent System:- A system consisting of two simultaneous linear equations is said to inconsistent, if it has no solution

The two equations- ax1 + by1 + c1 = 0 and ax2 + by2 + c2 = 0  are:  
(a) Parallel, if it has no solution
(b) Coincident, if they have infinite number of solutions
(c) Intersecting, if they have one solution  
Important points for exams:
1. Coordinate points in a plane:

• Let, A be a point in plane
• Let, m and n are the distance of A from x-axis and y-axis respectively
• Then, the coordinate of A will be (m,n).
• m is called x-coordinate or Abscissa of A
• n is called y-coordinate or Ordinate of A.

2. Coordinate point on x-axis: On X-axis, the y-coordinate or ordinate point will be Zero. Hence, the straight line will pass through x-coordinate only, parallel to y-axis. And, the coordinate of plane on x-axis will be (m,0)
3. Coordinate point on y-axis: On Y-axis, the x-coordinate or abscissa point will be Zero. Hence, the straight line will pass through y-coordinate only, parallel to x-axis. And, the coordinate of plane on y-axis will be (0,n)
4. The area bounded by |x| + |y| = k is 2k2.

For example: Find the area of |x| + |y| = 6cm. The answer will be 2x62 = 72 cm2.

How to draw Graph:

- Draw the graph of equation- 4x-y = 2.
- Now, putting x=0 and y=0 at a time to find the intercepts.
- i.e. x-intercept = 2/4 = 1/2 ;and y-intercept = -2

And, if putting x,y coordinates = 0, LHS=RHS then the shaded region will on the upper side of the line, otherwise it will be on the lower side of the line.