CHAPTER AT A GLANCE
System of a Pair of Linear Equations in Two Variables
An equations of the form Ax + By + C = 0 is called a linear equation in two variables x and y where A, B, C are real numbers.
Two linear equations in the same two variables are called a pair of linear equations in two variables. Standard form of linear equations in two variables.
Representation of Linear Equation in Two Variables
Every linear equation in two variables graphically represents a line and each solution (x, y) of a linear equation in two variables, ax+ by+ c = 0, corresponds to a point on the line representing the equation, and vice versa.
Ploting Linear Equation in Two Variables on the Graph
There are infinitely many solutions of each linear equation. So, we choose at least any two values of one variable & get the value of other variable by substitution. i.e; Consider; Ax +By+ C = 0
We can write the above linear equation as:
Here, we can choose any values of y & can find corresponding values of x. After getting the values of (x, y) we plot them on the graph thereby getting the line representing Ax +By+ C = 0.
Method of Solution of a Pair of Linear Equations in Two Variables
Coordinate of the point (x, y) which satisfy the system of pair of linear equations in two variables is the required solution. This is the point where the two lines representing the two equations intersect each other.
There are two methods of finding solution of a pair of Linear equations in two variables.
(1) Graphical Method: This method is less convenient when point representing the solution has non-integral solution.
(2) Algebraic Method : This method is more convenient when point representing the solution has non-integral co-ordinates.
This method is further divided into three methods:
(i) Substitution Method, (ii) Elimination Method and
(iii) Cross Multiplication Method.
Consistency and Nature of the Graphs
Consider the standard form of linear equations in two variables.
a1x + b1y + c1 = 0; a2x + biy + c2 = 0
While solving the above system of equation by this method following three cases arise.
Graphical Method of Solution
In this method, two equations are plotted separately in a single graph (as discussed in box-3).
Infinite set of solution, some of which are: Q(xl, y1) & R(x2, y2)
Algebraic Method of Solution
Consider the following system of equation
a1x+b1y+c1=0; a2x+b2y+c2 =0
There are three methods under the Algebraic method to solve the above system.
(I) Substitution method
- (a) Find the value of one variable, say yin terms of x or x in terms of y from one equation.
- (b) Substitute this value in second equation to get equation in one variable and find solution.
- (c) Now substitute the value/solution so obtained in step (b) in the equation got in step (a)
(ii) Elimination method
- (a) If coefficient of any one variable are not same in both the equation multiply both the equation with suitable non-zero constants to make coefficient of any one variable numerically equal.
- (b) Add or subtract the equations so obtained to get equation in one variable and solve it.
- (c) Now substitute the value of the variable got in the above step in either of the original equation to get value of the other variable.
(iii) Cross multiplication method
For the pair of Linear equations intwo variables: a1x+b1y+c1=O
a2x + b2y + c2 = 0
Consider the following diagram
Equations Reducible to a Pair of Linear Equations in Two Variables Sometimes pair of equaitons are not linear to start with, them they are altered so that they reduce to a pair of linear equations.
For Example:
Now we can use any method to solve them.
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