Linear equation – STD 09

General form of linear equation

ax + by + c = 0

where, a, b and c are real numbers.

a and b are not equal to zero.

x, y = two variables.

Image result for linear equation graph x axis and y axis

  • A linear equation in two variables has infinitely many solutions.
  • The graph of every linear equation in two variables is a straight line.

Image result for x = 0 graph

  • x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis.
  • The graph of x = a is a straight line parallel to the y-axis.

  • The graph of y = a is a straight line parallel to the x-axis.

Image result for The graph of x = a is a straight line parallel to the y-axis.

  • An equation of the type y = mx represents a line passing through the origin

Image result for the graph of y = mx passes through the origin

m = slope

Draw the lines of y = 2x and y =-2x

Image result for the graph of y = mx passes through the origin

Example Problems

Problem 1

 


Example 2

Solution of Linear Equation

a) Linear equation with a unique solution

b) Only two solutions

c) Infinitely many solutions

Answer:


Answer






Total fare = y

total distance = x

y = 1 x 8 + (x – 1) 5

y = 8 + 5x  – 5

y = 5x +3

5x -y +3 = 0

y = 5x +3

y(1) = 5 (1)  + 3 = 8

y(7) = 5(2) + 3 = 13

y(3) = 5(3) + 3 = 18

X 1 2 3
Y 8 13 18

NOTE:
Variable is classified into

  • independent variable (x)
  • dependent variable (y)




Draw the same graph in the third quadrant and find out at what point deg C = deg F


GRAPHS

Draw the graph of the line joining  the points (3, 5) and (-5, 1)

Steps:

1. Draw the x-axis and y-axis on a graph
sheet with 1 cm = 1 unit on both axes.
2. We plot the two given points (3, 5),
(-5, 1) on the graph sheet.
3. We join the points by a line segment
and extend it on either direction.
4. We get the required linear graph.


Draw the graph of y = 6x

Linear equation – general form: ax + by + c = 0

y = 6x

y = dependent variable; also represented as y = f(x)

x = independent variable.

6x – y + 0 = 0

a = 6; b = -1; c = 0


Draw the graph of x = 5

Since x is constant, it is parallel to y axis. So, the point may be (5, y). Y becomes independent variable. We may give any value to y such as -2, 0,  2.


 


y = 4x -1


 Draw the graph 2x + 3y = 12

A linear equation can also be represented as y = mx + c

where, m = slope

m  = Δy/Δx

c = intercept

Given equation

2x + 3y = 12

3y = -2x +12

y = 1/3 (-2x + 12)

y = -2x/3 + 12/3

y =(-2/3)x + 4

what is m?

m = -2/3

y intercept, c = (0, 4)

 

Image result for y = mx + c

Image result for y = mx + c

Is in form
y = mx + c
Ex 1 y = - ¾ x + 2
m = - ¾ , y int (0 , 2)
Ex 2 m = 9, y int (0 , -8)
y = 9x – 8
gradient Y intercep...

 

 

 


Problems for practice: