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Edit Comment Close Premium member Presentation Transcript Slide 1: linear equation in two variables “ The principal use of the analytic art is to bring mathematical problem to equations and to exhibit those equations in the most simple terms that can be .” Contents : Contents Introduction Linear equations Points for solving a linear equation Solution of a linear equation Graph of a linear equation in two variables Equations of lines parallel to x-axis and y-axis Examples and solutions summary Introduction : Introduction An excellent characteristic of equations in two variables is their adaptability to graphical analysis. The rectangular coordinate system is used in analyzing equations graphically. This system of horizontal and vertical lines, meeting each other at right angles and thus forming a rectangular grid, is often called the Cartesian coordinate system. Cartesian plane Introduction : Introduction A simple linear equation is an equality between two algebraic expressions involving an unknown value called the variable. In a linear equation the exponent of the variable is always equal to 1. The two sides of an equation are called Right Hand Side (RHS) and Left-Hand Side (LHS). They are written on either side of equal sign. LHS RHS 4x+3 = 15 2x+5y = 0 -2x+3y= 6 Slide 5: Introduction A linear equation in two variables is put in the form of ax+by+c=0,where a,b,c are real numbers, and a and b are not both zero. Equation 2x+3y=9 X+y/4-4=0 5=2x Y-2=0 2+x/3=0 A B C 3 -9 1/4 -4 0 5 0 1 -2 1/3 0 2 Linear equations : Linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can have one or more variables. Linear equations occur with great regularity in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state 5X+2=0 -5/2 -5 -4 -3 -2 -1 0 1 2 3 4 5 Points for solving a linear equation : Points for solving a linear equation the same is added to or subtracted from both the sides of equations . you multiply or divide both the sides of the equation by the same non-zero number. 5+y-2x=15 5+y-2x-5=15-5 y-2x=10 y-2x /2=10/2 Solution of a linear equation : Solution of a linear equation Every linear equation has a unique solution as there is a single variable in the equation to be solved but in a linear equation involving two variables in the equation, a solution means a pair of values, one for x and one for y which satisfy the given equation Example-p(x)=2x+3y (1)If x=3 2x+3y=(2x3)+(3xy)=12 6+3y=12 y=2, therefore the solution is (3,2) (2)If x=2 2x+3y=(2x2)+(3xy)=12 4+3y=12 y=8/3, therefore the solution is (2,8/3) Similarly many another solutions can be taken out from this single equation. That is ,a linear equation in two variables has infinitely many solutions. Graph of a linear equation in two variables : Graph of a linear equation in two variables Graph of a linear equation is representation of the linear equation geo. Observations on a graph- Every point whose coordinates satisfy the equation lies on the line AB. Every point on the line AB gives a solution of the equation. Any point, which does not lie on the line AB is not a solution of equation. X+2Y=6 Graph of a linear equation in two variables : Graph of a linear equation in two variables EXAMPLES - Graph for the equation-x+y-2=0 If y=2;x=0 If y=4;x=-2 If y=3;x=-1 If y=1;x=1 Equations of lines parallel to x-axis : Equations of lines parallel to x-axis The graph of y=a is a straight line parallel to the x-axis y=4 2y-7=1 2y-7+7=1+7 2y=8 2y/2=8/2 y=4 x y (2y-7=1) Slide 12: Equations of lines parallel to y-axis x y The graph of x=a is a straight line parallel to the y-axis 3x-10=5 3x=15 x=5 x=5 (3x-10=5) Examples and solutions : Examples and solutions Give the values of a, b and c : -2x+3y=9 a=-2 b=3 c=-9 5x-3y=-4 a=5 b=-3 c=4 3x+2=0 a=3 b=0 c=2 Y-5=0 a=0 b=1 c=-5 Examples and solutions : Examples and solutions Write 2 solutions for each: X+2y=6 If y=1;x=4 If y=2;x=2 2x+y=4 If x=1;y=2 If x=2;y=0 4x-2y=6 If x=1;y=-1 If x=2;y=1 Examples and solutions : Examples and solutions Draw the graph of the equation: 2+2y=6x If x=2;y=5 If x=1;y=2 If x=0;y=-1 (1,2) (0,-1) (2,5) 2+2y=6x Examples and solutions : Give the geometric representation of 2x+8=0 as an equation in two variables: Examples and solutions y=-4 x y (2x+8=0) (-4,3) (-4,-3) SUMMARY : SUMMARY An equation of the form ax+by+c=0,wherea,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 variables has infinitely many solutions. The graph of every linear equation in two variables is a straight line. 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. 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. Moreover, every solution of the linear equation is a point on the graph of the linear equation. Slide 18: COMPILED BY- GROUP D SUBMITTED TO- L.R.SHARMA You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.