1. An equation which can be put in the form , where and are real numbers and and are not both zero, is called a linear equation in two variables and .
2. Every solution of the equation is a point on the line representing it. Or each solution , of a linear equation in two variables . corresponds to a point on the line representing the equation and vice-versa.
3. A linear equation in two variables has an infinite number of solutions.
4. a pair of such equations is called a system of linear equations.
5. We have three types of systems of two linear equations.
(i) Independent System, which has a unique solution. Such system is termed as a consistent system with unique solution.
(ii) Inconsistent system, which has no solution.
(iii) Dependent System, which represents a pair of equivalent equations and has an infinite number of solution. Such system with infinite solution.
6. A pair of linear equations in two variables which has a common point, i.e., which has only one solution is called a consistent pair of linear equations.
7. A pair of linear equations in two variables which has no solution, i.e., the lines are parallel to each other is called an inconsistent pair of linear equations.
8. Pair of linear equations in two variables which are equivalent and has infinitely many solutions are called dependent pair of linear equations. Note that a dependent pair of linear equations is always consistent with infinite number of solutions.
9. A pair of linear equations and represents
(i) intersection lines, then
(ii) parallel lines, then
(iii) coincident lines, then
converse of the above statement is also true.
10. Graphical Method of Solving a Pair of Linear Equations
(a) To solve a system of two linear equations graphically :
(i) Draw graph of the first equation.
(ii) On the same pair of axes, draw graph of the second equation.
(b) After representing a pair of linear equations graphically, only one of the following three possibilities can happen :
(i) The two lines will intersect at a point.
(ii) The two lines will be parallel.
(iii) The two lines will be coincident.
(c) (i) If the two lines intersect at a point, read the coordinates of the point of intersection to obtain the solution and verify
(ii) If the two lines are parallel, i.e., there is no point of intersection, write the system as inconsistent. Hence, no solution.
(iii) If the two lines have the same graph, then write the system as consistent with infinite number of solutions.
11. Algebraic Methods of Solving a Pair of Linear Equations
(a) Substitution Method :
(i) Suppose we are given two linear equations in and . For solving these equations by the substitution method, we proceed according to the following steps :
Step 1. Express in terms of in one of the given equations.
Step 2. Substitute this value of in terms of in the other equation. This gives a linear equation in .
Step 3. Solve the linear equation in obtained in step 2.
Step 4. Substitute this value of in the relation taken in step 1 to obtain a linear equation in .
Step 5. Solve the above linear equation in to get the value of .
Note : We may interchange the role of and in the above method.
(ii) While solving a pair of linear equations, if we get statements with no variables, we conclude as below.
(a) If the statement is true, we say that the equations have infinitely many solutions.
(b) If the statement is false, we say that the equations have no solution.
(iii) When the two given equations in and are such that the coefficients of and in one equation are interchanged in the other, then we add and subtract the two equations to get a pair of very simple equations.
(b) Elimination Method : In this method, we eliminate one of the variables and proceed using the following steps.
Step 1. Multiply the given equations by suitable numbers so as to make the coefficients of one of the variables equal.
Step 2. If the equal coefficients are opposite in sign, then add the new equations Otherwise, subtract them.
Step 3.The resulting equations is linear in one variable.
Solve it to get the value of one of the unknown quantities.
Step 4. Substitute this value in any of the given equations.
Step 5. Solve it to get the value of the other variable.
(c) Cross Multiplication Method :
(i) The system of two linear equations
where has a unique solution, given by
We generally write it as
following diagram will help to apply the cross multiplication method directly.
The arrows between the numbers indicate that they are to be multiplied. The products with upward arrows are to be subtracted from the products with downward arrows.
(ii) The system of equations
(a) is consistent with unique solution, if ,
i.e., lines represented by equations (i) and (ii) intersect at a point
(b) is inconsistent , if , i.e., lines represented by equations (i) and (ii) are parallel and non coincident.