Linear Equations in Two Variables
Linear Equations in Two Variables
An equation which contains two variables and the degree
of each term containing variable is one, is called a linear equation in two
variables.
ax + by + c = 0 is the general form of a linear equation in two
variables;
a, b, c are real numbers and a, b are not equal to zero at the same time.
Solve the following simultaneous equations.
(1) 5x - 3y = 8; 3x + y = 2 2) 3x + 2y = 29; 5x - y = 18
3) 15x + 17y = 21; 17x + 15y = 11
Solutions
1) 5x - 3y = 8; 3x + y = 2
5x - 3y = 8. . . (I)
3x + y = 2 . . . (II)
Multiplying both sides of equation (II) by 3 we get question
(III)
9x + 3y = 6 . . . (III)
5x - 3y = 8. . . (I)
Now let us add equations (I) and (III)
Substituting x = 1 in equation (II)
3x + y = 2
∴ 3 × 1 + y = 2
∴
3 + y = 2
∴
y = -1
Solution is x = 1 and y = -1; it is also written as (x, y) = (1, -1)
2) 3x + 2y = 29; 5x - y = 1
3x + 2y = 29. . . (I)
5x - y = 18 . . . (II)
Let’s solve the equations by eliminating ’y’.
Multiplying equation (II) by 2 we get question (III)
∴ 5x × 2 – y × 2 = 18 × 2
∴ 10x - 2y = 36 …….. (III)
Let’s add equations (I) and (III)
3x + 2y = 29
+ 10x - 2y = 36
________________
13 x = 65
∴
x = 5
Substituting x = 5 in equation (I)
3x + 2y = 29
∴ 3 × 5 + 2y = 29
∴
15 + 2y = 29
∴ 2y = 29 – 15
∴ 2y = 14
∴
y = 7
Solution is x = 5 and y = 7; it is also written as (x, y) = (5, 7)
3) 15x + 17y = 21; 17x + 15y = 11
15x + 17y = 21……….. (I)
17x + 15y = 11……...... (II)
In the two equations above, the coefficients of x and y are interchanged. While solving such
equations we get two simple equations by adding and subtracting the given
equations. After solving these equations, we can easily find the solution.
Let’s add the two given equations.
Dividing both sides of the equation by 32 we get equation
(III)
x + y = 1 . . . (III)
Now, let’s subtract equation (II) from (I)
Dividing the equation by 2 we get equation (IV)
-x + y = 5 . . . (IV)
Now let’s add equations (III) and (IV).
∴ y = 3
x + y = 1
∴ x + 3 = 1
∴
x = 1 - 3
∴ x = -2
Solution is x = -2 and y = 3; it is also written as (x, y) = (-2, 3)
For practice
Solve the equations.
(1) x + y = 6 ; x - y = 4
(2) x + y = 5 ; x - y = 3
(3) x + y = 0 ; 2x - y = 9
(4) 3x - y = 2 ; 2x - y = 3
(5) 3x - 4y = -7 ; 5x - 2y = 0
Determinant
Degree of this determinant is 2, because there are 2
elements in each column and 2 elements in each row. Determinant represents a
number which is (ad-bc).
Determinants, usually, are represented with capital
letters as
A, B, C, D . . . etc.
Find the values of the following determinants.
For Practice
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