LINEAR EQUATION IN TWO VARIABLES
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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.
LINEAR EQUATION IN TWO VARIABLES
- 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 . ------------------------------------------------------------------ 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