Linear Equations
An equation is said to be linear if
the degree of the variable is 1. For example, x + 7 = 0, 3x + 4y = 12, x + y =
5 are linear equations.
Linear Equation in Two Variables
An
equation of the form ax + by = c, where a, b, c, are constants and a, b ≠ 0 is
a linear equation in two variables x and y.
Simultaneous Linear Equations
Two linear equations in two variables
which are satisfied by the same pair of values of the variables are called
simultaneous linear equations in two variables.
For example, 2x – y = 7 and 3x – 2y =
9.
The values of x and y which satisfy
both the equations are called the solutions of the simultaneous linear
equations. Here, in the above example if we put x = 5 and y = 3, we observe
that both the equations are satisfied. Thus, we say that x = 5 and y = 3 is the
solution set of the given equations.
Note that the solution set of the pair
of simultaneous linear equations is unique.
Methods of Solving Simultaneous Linear Equations
There are
three methods for solving a pair of simultaneous linear equations in two
variables:
1.
Substitution Method
2. Elimination
(or AdditionSubtraction) Method
3.
Comparison Method
Substitution Method
In order to
solve a pair of simultaneous linear equations in two variables, proceed as
follows:
1. Find the
value of one of the variable by solving any one of the given equations.
2.
Substitute that value of the variable in the other equation.
3. Now,
solve the required linear equation in one variable and substitute this value
into either of the two original equations and solve it to find the value of the
other variable.
Example: Solve the following pair of
simultaneous linear equations:
3x + 5y
= 28
x – 7y
= –34
Solution:
Given equations are
3x + 5y = 28 … (1)
x – 7y = –34 … (2)
Let us first find the value of one of
the variable from any one of the given equations. In order to avoid fractions,
let us express x in terms of y using the equation (2)
i.e., x – 7y = –34 ⇒
x = –34 + 7y
Now, putting x = –34 + 7y in equation
(1), we get 3 × (−34 + 7y) + 5y = 28
⇒ –102 + 21y +
5y = 28 ⇒
26y = 130
⇒ y = 130/26 ⇒ y = 5
Now, putting this value of y in
equation (2), we have x – 7 × 5 = −34 ⇒ x = −34 + 35 = 1
∴ the solution set is x = 1 and y = 5.
Elimination or AdditionSubtraction Method
In order to
avoid substituting fractions in solving equations, we use elimination method.
To solve the pair of simultaneous linear equations, proceed as follows:
1. Transform
the equations by multiplying one or both the equations to eliminate one
variable by adding or subtracting.
2. Now,
solve the required linear equation in one variable and substitute this value
into either of the two original equations and solve it to find the value of the
other variable.
Example: Solve: 3x + 4y = 11 ... (1)
x + 7y = 15
... (2)
Solution: If we multiply equation (2) by 3, we shall
have the same coefficient of x in both equations.
The new equation is 3x + 21y = 45 ... (3)
We can now eliminate x by taking away,
i.e., subtracting (1) from equation (3)
3x + 21y = 45 ... (3)
3x + 4y = 11 ... (1)
17y = 34 Þ y
= 2
We substitute y = 2 in equation (2),
we have x + 14 = 15 Þ x = 1.
Hence, the solutions are x = 1 and y =
2.
To check these values, substitute them
in equation (1), we have
L.H.S. = 3 ´ 1 + 4 ´
2 3 + 8 = 11 = R.H.S.
As the L.H.S. = R.H.S. the solutions
are correct.
Comparison Method
In order to
solve a pair of simultaneous linear equations in two variables, proceed as
follows:
1. Transform both the equations in terms
of one of the same variable.
2. Equate the values so obtained and
then solve the required linear equation.
3. Substitute the value of the variable
in either of the two equations and solve it to fi nd the value of the other
variable.
Example: Solve the following pair of equations.
2x + 3y = 2 …
(1)
4x + 5y = 6 …
(2)
Solution: Given equations are:
2x + 3y = 2 …
(1)
4x + 5y = 6 … (2)
Expressing
equation (1) and (2) in terms of one variable say x, we get
x = (2 – 3y)/2 … (3) and x = (6 – 5y)/4 … (4)
Now,
equating equation (3) and (4) and then solving, we get
(2 – 3y)/2 =
(6 – 5y)/4
⇒ 2(2 – 3y) = 6 – 5y ⇒ 4 – 6y = 6 – 5y
⇒ –6y + 5y = 6 – 4 ⇒ –y = 2 ⇒ y = –2
Putting y =
–2 in equation (3), we get x = 4
∴ the solution is x = 4 and y = –2.
Two more
methods are also used to solve the simultaneous linear equations in one
variable.
1. Graphical method
2. Crossmultiplication method
Graphical Method of Solving Simultaneous Linear Equations
Example: Solve by
the graphical method: x + 3y = 6, 2x – 3y = 12.
Solution: From first equation, y = (6 – x)/3
x

0

6

y = (6 – x)/3

2

0

So, the points are (0, 2) and (6, 0).
From second equation, y = (2x – 12)/3
x

0

3

y = (2x – 12)/3

4

2

So, the points are (0, 4) and (3, 2).
Now, plot these points for both the
equations on the same graph paper.
From
the graph, we find the common point of intersection, i.e., (6, 0).
So,
the solution of the equations is (6, 0).
Crossmultiplication Method of Solving Simultaneous Linear Equations
The
general form of a pair of linear equations in two variables is:
a_{1}x_{1} + b_{1}y_{1} + c_{1
}= 0 … (1)
a_{2}x_{2} + b_{2}y_{2} + c_{2
}= 0_{ } … (2)
Write the coefficients
of x_{1}, y_{1}, x_{2} and y_{2} as follows:
b_{1} c_{1} a_{1} b_{1}
b_{2} c_{2} a_{2} b_{2}
Now, cross multiply and
write as follows:
x/(b_{1}c_{2}
– b_{2}c_{1}) = y/(c_{1}a_{2} – c_{2}a_{1})
= 1/(a_{1}b_{2} – a_{2}b_{1})
Now, x = (b_{1}c_{2}
– b_{2}c_{1})/ (a_{1}b_{2} – a_{2}b_{1})
and y = (c_{1}a_{2} – c_{2}a_{1})/ (a_{1}b_{2}
– a_{2}b_{1})
Example: Solve for x
and y by crossmultiplication method:
2x – y = 6 and x – y = 2
2x – y = 6 and x – y = 2
Nature of Simultaneous Linear Equations
The
general form of a pair of linear equations in two variables is:
a_{1}x_{1} + b_{1}y_{1} + c_{1
}= 0 … (1)
a_{2}x_{2} + b_{2}y_{2} + c_{2
}= 0_{ } … (2)
There are three
conditions:
1.
If a_{1}/ a_{2 }≠ b_{1}/ b_{2},
then both the equations have a unique solution and they are consistent.
2.
If a_{1}/ a_{2 }= b_{1}/ b_{2
}=c_{1}/c_{2}, then both the equations have infinitely
many solutions and they are dependent and consistent.
3.
If a_{1}/ a_{2 }= b_{1}/ b_{2
}≠ c_{1}/c_{2}, then both the equations have no solution
and the equations are said to be inconsistent.