Introduction:
We are already familiar with solving linear equations of one variable. Let us discuss the situation as follows. 4 men and 4 boys can do a piece of work in 3 days while 2 men and 5 boys can finish it in 4 days. How long would it take 1 boy to do it? How long would it take 1 man to do it? .
The above situation seems to be hard to solve. But using the method of solving simultaneous equations of two variables we can solve it.
In this topic let us discuss with the linear equations with 2 variables, types of solutions and the methods of solving the linear equations in 2 variables.
Systems of linear equations in two variables :
a_{1} x + b_{1} y = c_{1} and
a_{2} x + b_{2} y = c_{2}
The above system of linear equations of two variables satisfy any one of the following conditions.
1. One solution.
2. No solution
3. Infinitely many solutions
The above system of equations are also called simultaneous linear equations in two variables.
How to solve linear equations in two variables
1. Substitution Method.
2. Elimination Method.
3. Cross multiplication Method
4.. Graphical Method
5. Matrix Method
6. Method of Determinants (Cramer's Rule)
and many more methods
How to do linear Equations in two variables:
The linear equations in two variables can be solved by any one of the above methods. We should make sure that we frame the equations according to the standard form, ax + by + c = 0
Solving System Linear Equations in Two Variables
The graph of the linear equations is a straight line.
When we graph the lines on a graph, one of the following conditions will be satisfied.
1. If the lines intersect at a point there will be one and only one solution. The solution is written in the form (x,y).
2. If the lines are parallel there will be no solution, as the lines do not meet.
3. If the lines coincide there will be infinite number of solutions.
Consistency: We say that the system of equations is consistent if it has at least one solution.
Intersecting lines are coinciding lines are the consistent system equations.
Inconsistency: We say that the system of equations is inconsistent if it has no solution.
Parallel lines are inconsistent system of equations.
Linear equations and inequalities in two variables
Linear Equations in Two Variables:
When we consider the linear equations in two variables, their solutions will satisfy any one of the following conditions.
1. Only one solution (unique solutions)
2. No solution
3. Infinitely many solutions satisfying the given equation.
we shall discuss the more examples of linear equations in two variables in this section.
Linear Inequalities in two variables:
When we consider the linear inequalities in two variables, the expressions and the constants are related by the sign >, <, <= or > =.
Let us consider the following inequalities.
x+ y <= 4 and 2x + 3y > = 9
The region common to both the shaded region (checked in black and red ) is the solution to the above inequality.
system of linear equations in two variables
Linear Equations in two variables examples:
Solve : 2x + 3y = 11 ; x  2y = 2 by the Method of Substitution.
Solution: By the method of substitution we find one of the variable from one of the equation and substitute it in the other equation.
We have, 2x  3 y = 11 (1)
and x  2y = 2 (2)
x  2y = 2
=> x = 2 + 2y
Substituting x = 2  2y in equation (1), we get
2 ( 2 + 2y ) + 3y = 11
=> 4 + 4y + 3y = 11
=> 4 + 7y = 11
=> 7y = 11  4 = 7
=> y = 7/7 = 1
Substituting y = 1, in x = 2 + 2y, we get
x = 2 + 2(1) = 2 + 2= 4
Hence the solution to the above equation is ( x, y ) = ( 4, 1)
Linear Equations in two variables formula:
Cross Multiplication Method:
Consider the two equations,
a_{1} x + b_{1} y = c_{1} and
a_{2} x + b_{2} y = c_{2}The solution to the above two equations is given by,
$x = \frac{b_{1}c_{2} b_{2}c_{1}}{a_{1}b_{2} b_{2}a_{1}}$
$y = \frac{c_{1}a_{2} c_{2}a_{1}}{a_{1}b_{2} b_{2}a_{1}} $
By comparing the given equations to these standard equations we can find the variables, a_{1}, a_{2}, b_{1}, b_{2}, c_{1} , c_{2, }and then
substitute in the above equation.
Examples of linear equations in two variables
Step 1 : xy =5 implies x=5+y.
Step 2 : Put this in equation 2 that is x+y=4
5+y+y=4
5+2y=4
2y=1
y=1/2.
Step 3 : x(1/2)=5
x=51/2
x=9/2.
Solution =9/2.
Linear equations in two variables word problems
Example : The monthly incomes of Kevin and Keydon are in the ratio 3:4 and their monthly expenditures are in the ratio 5 : 7. If each saves 2,500 dollars per month. Find their monthly incomes.
Solution: Let us assume that the monthly incomes expenditures as per the table given below.
Kevin  Keydon 

Income 
3x  4x 
Expenditure  5y  7y 
Savings  3x5y  4x7y 
Equations  3x5y = 2500  4x  7y = 2500 
4x  7y = 2500 (2)
Multiplying (1) by 4, and (2) by 3, we get,
4 ( 3x  5y ) = 4(2500)
=> 12 x  20 y = 10000 (3)
3(4x  7y) = 3 ( 2500)
=> 12 x  21 y = 7500 (4)
Subtracting (3) and (4), we get,
12 x  20 y  [ 12 x  21 y ] = 10000  7500
=> 12 x  20 y  12 x + 21 y = 2500
=> y = 2500
Substituting y = 2500 in (1), we get,
3x  5y = 2500
=> 3x  5(2500) = 2500
=> 3x  12500 = 2500
=> 3x = 12500+ 2500 = 15,000
=> x = 15000/3 = 5000
From the above table, we have Income of Kevin = 3x = 3 ( 5000) = 15,000 dollars
Income of Keydon = 4x = 4( 5000) = 20,000 dollars
Practice Questions:
Solve the following system of linear equations in two variables.
1. 3x  4y = 7 ; x + y = 9 by substitution method.2. $\frac{2}{x}+\frac{3}{y}= 17$ ; $\frac{1}{x}+ \frac{1}{y}= 7$.
3. If the length and breadth of a room are increased by 1m each, its area is increased by 21 m^{2} . If the length is increased by 3 m and breadth decreased
by 1 m, the area is decreased by 5 m^{2}. Find (i) the dimensions of the room (ii) the area of the room.
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