1.
The equation of the form ax = b or ax + b = 0, where a and b are two real numbers such that x ≠ 0
and x is a variable is called a inear equation in one variable.
2. The general form of a linear equation in two variables is ax + by +c = 0 or ax + by = c where a, b, c
are real numbers and a ≠ 0, b ≠ 0 and x, y are variables.
3. The graph of a linear equation in two variables is a straight line.
4. The graph of a linear equation in one variable is a straight line parallel to x-axis for ay = b and
parallel to y-axis for ax = b, where a ≠ 0.
5. A pair of linear equations in two variables is said to form a system of simultaneous linear
equations.
6. The value of the variable x and y satisfying each one of the equations in a given system of linear
equations in x and y simultaneously is called a solution of the system.
Standard form of linear equation:
(Standard form refers to all positive coefficient)
a1x + b1y + c1 = 0 ....(i)
a2x + b2y + c2 = 0 ....(ii)
For solving such equations we have three methods.
(i) Elimination by substitution
(ii) Elimination by equating the coefficients
(iii) Elimination by cross multiplication.
ELIMINATION BY SUBSTITUTION:
(Find the value of any one variable in terms of other and then use it to find other variable from the
second equation):
Linear Equations In Two Variables
question 1. Solve x + 4y = 14 .....(i)
7x - 3y = 5 ....(ii)
Solution: From equation (i) x = 14 - 4y ....(iii)
Substitute the value of x in equation (ii)
⇒ 7 (14 - 4y) - 3y = 5
⇒ 98 - 28y - 3y = 5
⇒ 98 - 31y = 5
⇒ 93 = 31y
⇒ y = 93/31
⇒y=3
Now substitute value of y in equation (iii)
⇒ 7x - 3 (3) = 5
⇒ 7x - 3 (3) = 5
⇒ 7x = 14
⇒ x = 14/7 = 2
So the solution is x = 2 and y = 3
ELIMINATION BY EQUATING THE COEFFICIENTS:
(Eliminating One variable by making the coefficient equal to get the value of one variable and then
put it in any equation to find other variable):
Linear Equations In Two Variables
question 1. Solve 9x - 4y = 8 ...(i)
13x + 7y = 101 ...(ii)
Solution: Multiply equation (i) by 7 and equation (ii) by 4, we get
adding 63x – 28y = 56
52x + 28y = 404
115x = 460
⇒ .x = 460/115
⇒x=4
Substitute x = 4 in equation (i)
9 (4) –0 4y = 8 ⇒ 36 – 8 = 4y ⇒ 28 = 4y
⇒ y = 28/4 = 7
So the solution is x = 4 and y = 7.
COMPARISON METHOD:
(Find the value of one variable from both the equation and equate them to get the value of other
variable):
Let any pair of linear equations in two variables is of the form
a1x + b1y + c1 = 0 .........(i)
a2x + b2y + c2 = 0 .....…(ii)
Linear Equations In Two Variables
question 1. Solve for x and y : 2x + 3y = 8 , x - 5y = -9.
Solution: 2x + 3y = 8 …(i)
x - 5y = -9. …(ii)
From equation (i),
2x = 8 - 3y , x = 8 - 3y/2 …(iii)
From equation (ii),
x = -9 + 5y…(iv)
From equation (iii) and (iv), we have
Linear Equations In Two Variables
Putting value of y in equation (iv), we have
x = 8 - 3 x 2/2
x=1
Solution x = 1, y = 2.
CROSS MULTIPLICATION METHOD:
Let the equation
a1x + b1y + c1 = 0 …(i)
a2x + b2y + c2 = 0 …(ii)
To obtain the values of x and y, we follow these steps:
1. Multiply Equation (i) by b2 and (ii) by b1, we get
Linear Equations In Two Variables
2. Subtracting Equation (iv) from (iii), we get:
Linear Equations In Two Variables
3. Substituting this value of x in (i) or (ii), we get
Linear Equations In Two Variables. …(vi)
We can write the solution given by equations (v) and (vi) in the following form:
Linear Equations In Two Variables …(vii)
In remembering the above result, the following diagram may be helpful:
Linear Equations In Two Variables
For solving a pair of linear equations by this method, we will follow the following steps:
1. Write the given equations in the form (i) and (ii).
2. Taking the help of the diagram above, write equations as given in (viii).
3. Find x and y.
Graphic3.jpg
For the equations like
a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 The solution set can be calculated by using
Linear Equations In Two Variables
(Please note position of c1 and c2 with equality sign and subsequent change in the third term i.e. –1)
question 1. Solve the following system of equations in x and y
ax + by - a + b = 0, bx - ay - a - b = 0
Solution: a1 = a, b1 = b, a2 = b and b2 = –a
∴ Linear Equations In Two Variables
Hence, the given system of the equation has unique solution.
By cross multiplication, we have
Linear Equations In Two Variables
or x = 1 and y = –1
question 2. Solve for x and y.
Linear Equations In Two Variables
Solution: Here a1 =1/a, b1 =1/b, a2 = 1/a2 and b2 = 1/b2
∴ Linear Equations In Two Variables.
Hence, the given system of equation has unique solution.
Linear Equations In Two Variables
By cross multiplication, we have
Linear Equations In Two Variables
question 3. Solve the following system of
