WHAT IS LINEAR EQUATION?
An equation between two variables that gives a straight line when plotted on a graph.
WHAT IS LINEAR EQUATION IN TWO VARIABLE?A linear equation in two variables is an equation with two variables (usually called x and y) where the variables are at most multiplied by a number, and added to something else. No exponents, no variables in denominators, no fancy functions of the variables.For example 2x-y=3 and x+y=3 are linear equations.
TYPES OF SOLUTIONS OF SYSTEMS OF EQUATIONS
• One solution – the lines cross at one point
• No solution – the lines do not cross
• Infinitely many solutions – the lines coincide
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES CAN BE SOLVED BY THE:
(I) GRAPHICALLY METHOD
(II) ALGEBRAIC METHOD
TO FIND THE VALUES OF X AND Y BY ALGEBRAIC METHOD
THERE ARE THREE METHODSSubstitution MethodElimination MethodCross-Multiplication Method
We can also find the values of X and Y graphically by finding there co-ordinates and then plotting on the graph
FINDING THE VALUES OF X AND Y BY GRAPHICAL METHOD
• We can also find the values of X and Y graphically by finding there co-ordinates and then plotting on the graph
SUBSTITUTION METHODSTEPS
Obtain the two equations. Let the equations be
a1x + b1y + c1 = 0 ----------- (i)
a2x + b2y + c2 = 0 ----------- (ii)
Choose either of the two equations, say (i) and find the value of one variable , say ‘y’ in terms of x
Substitute the value of y, obtained in the previous step in equation (ii) to get an equation in x
Solve the equation obtained in the previous step to get the value of x. Substitute the value of x and get the value of y.
• The method of substitution is not preferable if none of the coefficients of x and y are 1 or -1. For example, substitution is not the preferred method for the system below: 2x – 7y = 3
-5x + 3y = 7 • A better method is elimination by addition. The following operations can
be used to produce equivalent systems: • 1. Two equations can be interchanged. • 2. An equation can be multiplied by a non-zero constant. • 3. An equation can be multiplied by a non-zero constant and then
added to another equation
ELIMINATION METHOD
CROSS-MULTIPLICATION METHOD
• Let’s consider the general form of a pair of linear equations.
1++1=0 ++=0
To solve this pair of equations for and using cross-multiplication, we’ll arrange the variables and their coefficients
1, and , and the constants 1,
We can convert non linear equations in to linear equation by a suitable substitution
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