This document provides information about solving pairs of linear equations in two variables. It discusses several methods for solving such equations, including elimination, substitution, and cross-multiplication. The elimination method involves multiplying equations by constants to eliminate one variable, then solving the resulting equation. The substitution method finds one variable in terms of the other from one equation, substitutes it into the other equation, and solves. Cross-multiplication uses the formula that the ratio of the coefficients of the variables in one equation equals the ratio of the constants. Word problems can also be solved by setting up linear equations from the information provided.

1. PAIR OF LINEAR
EQUATIONS IN
TWO
VARIABLES
Class 10

2. Introduction • A common form of linear equation in the two variables
x and y is y=ax+c where a and c are constants.
• In this equation, the a determines slope and c
determines point at which line crosses the y axis.
In this, (3,2) is the solution of
pair of linear equations as
graph for each equation cuts
at point (3,2)

3. About Linear
Equations
• Linear Equation - 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 variables has infinitely many solutions.
• The graph of every linear equation in two variables is a
straight line.
• 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.
• 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.

4. Elimination
method
• Step 1 : First multiply both the equations by some suitable nonzero constants to make the
coefficients of one variable (either x or y) numerically equal.
• Step 2 :Then add or subtract one equation from the other so that one variable gets eliminated.
If you get an equation in one variable, go to
• Step 3. If in Step 2, we obtain a true statement involving no variable, then the original pair of
equations has infinitely many solutions. If in Step 2, we obtain a false statement involving no
variable, then the original pair of equations has no solution, i.e., it is inconsistent. Step 3 : Solve
the equation in one variable (x or y) so obtained to get its value.
• Step 4 : Substitute this value of x(or y) in either of the original equations to get the value of the
other variable.

6. Substitution
Method
• Step 1 : Find the value of one variable, say yin terms of
the other variable, i.e., x from either equation,
whichever is convenient.
• Step 2 : Substitute this value of y in the other equation,
and reduce it to an equation in one variable, i.e., in
terms of x, which can be solved. Sometimes, as in
Examples , you can get statements with no variable. If
this statement is true, you can conclude that the pair of
linear equations has infinitely many solutions. If the
statement is false, then the pair of linear equations is
inconsistent.
• Step 3 :Substitute the value of x(or y) obtained in Step
2 in the equation used in Step 1 to obtain the value of
the other variable.

8. Cross-
Multiplicatio
n method
• The formula for cross-multiplication and its use in
solving two simultaneous equations can be presented
as:
If (a1b2 - a2b1) ≠ 0 from the two simultaneous linear
equations
a1x + b1y + c1 = 0 ----------- (i)
a2x + b2y + c2 = 0 ----------- (ii)
We get, by the cross-multiplication method:
x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(a1b2 - a2b1)

10. Word
Problems
• Word problems based on linear equations in 2 variables
can easily be solved by the following this method.
• One number is 28 more than three times another number.
If each number were multiplied by four, their difference
would be 232.What are the numbers?
Solution:
a =3b +28
4a - 4b = 232
a = ? b = ?
4a - 4b = 232
4(3b +28) - 4b = 232
12b + 112 - 4b = 232
8b + 112 = 232
8b = 232 - 112
8b = 120
b = 120/8 = 15 =>b = 15
a = 3b +28 = 3(15) + 28 = 45 + 28 = 73 => a =73

11. Thank You • Hope you Enjoyed
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