This document discusses linear equations and their properties and applications. It defines a linear equation as one where each term is either a constant or the product of a constant and two variables. Linear equations can be represented as ax + by + c = 0 and their graphs are straight lines. The solutions of a linear equation are the points that satisfy the equation. Linear equations are used to model many real-world situations where a change in input results in proportional change in output, such as doubling recipes, calculating grass growth rates, and budgeting money for various tasks. While useful for modeling within a "linear regime," systems often become nonlinear if inputs are increased too much.

2. Introduction
A linear equation is an algebraic equation in
which each term is either a constant or the
product of a constant and two variables.
An equation of the form ax +by + c = 0 where a ,
b , and c are real numbers ,such that a and b are
not both zero, is called a linear equation in two
variables.
A linear equation in two variable has infinitely
many solution.
The graph of every linear equation in two
variables is a straight line.

3. X=0 is the equation of the y-axis and y=0
is the equation of x-axis.
The graph of x=0 is a straight line parallel
to y-axis.
The graph of y=0 is a straight line parallel
to the x-axis.
An equation of the type y=mx represents
a line passing through the origin.
Every point on the graph of a linear
equation in two variables is a solution of
the linear equation .

4. How its obtain?
The solutions of a linear equation can be
obtained by substituting different values for x in
the equation to find the corresponding values of y.
The values of x and y are represented as an order
pair. To plot the graph of a linear equation, its
solutions are found algebraically and then the
points are plotted on the graph.
Any linear equation of the form 'ax + by + c = 0'
represents a straight line on the graph. The points
of the straight line make up the collection of
solutions of the equation.

6. Algorithm
Obtain the linear equation . Let the equation
the equation be ax + by + c=0.
Give any three values to x and calculate the
corresponding values of y to obtain solutions .
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If possible ,choose integral values of x in such a
way that the corresponding values of y are also
integers.

7. Equation-
3x-2y=6
Example
X Y
2 0
4 3
6 6

8. Examples

9. Linear Equation in real
life
One of the realities of life is how so much of the
world runs by mathematical rules. As one of
the tools of mathematics, linear systems have
multiple uses in the real world. Life is full of
situations when the output of a system
doubles if the input doubles, and the output
cuts in half if the input does the same. That's
what a linear system is, and any linear system
can be described with a linear equation.

10. Example 1
If you've ever doubled a favorite recipe, you've
applied a linear equation. If one cake equals
1/2 cup of butter, 2 cups of flour, 1 tsp. of
baking powder, three eggs and 1 cup of sugar
and milk, then two cakes equal 1 cup of
butter, 4 cups of flour, 2tsp. of baking
powder, six eggs and 2 cups of sugar and
milk. To get twice the output, you put in
twice the input. You might not have known
you were using a linear equation, but that's
exactly what you did.

11. Example 2
SAM has also noticed that it's springtime. The
grass has been growing. It grew 2 inches in
two weeks. He doesn't like the grass to be
taller than 2 1/2 inches, but he doesn't like to
cut it shorter than 1 3/4 inches. How often does
he need to cut the lawn? He just puts that
calculation in his linear expression, where (14
days/2 inches) * 3/4 inch tells him he needs to
cut his lawn every 5 1/4 days. He just ignores
the 1/4 and figures he'll cut the lawn every five
days.

12. Where they are…
It's not hard to see other similar situations. If
you want to buy coke for the big party and
you've got 360Rs. in your pocket, a linear
equation tells you how much you can afford.
Whether you need to bring in enough wood for
the fire to burn overnight, calculate your
paycheck, figure out how much paint you need
to redo the upstairs bedrooms or buy enough
petrol to make it to and from your Mausi’s
house, linear equations provide the answers.
Linear systems are, literally, everywhere.

13. Where they are not…
One of the paradoxes is that just about every linear
system is also a nonlinear system. Thinking you can
make one giant cake by quadrupling a recipe will
probably not work. If there's a really heavy snowfall
year and snow gets pushed up against the walls of
the valley, the water company's estimate of available
water will be off. After the pool is full and starts
washing over the edge, the water won't get any
deeper. So most linear systems have a "linear regime"
--- a region over which the linear rules apply--- and a
"nonlinear regime" --- where they don't. As long as
you're in the linear regime, the linear equations hold
true.

14. Thank you