1.Linear Equations in Two Variables
Practice Set 1.1
1.Linear Equations in Two Variables
(1) Complete the following activity to solve the simultaneous equations.
5x + 3y = 9 —–(I)
2x – 3y = 12 —– (II)
Solution:
5x + 3y = 9 —–(I)
2x – 3y = 12 —– (II)
Let’s add equations (1) and (2),
1.Linear Equations in Two Variables
2. Solve the following simultaneous equations.
(1) 3a + 5b = 26; a + 5b = 22
Solution:
3a + 5b = 26 (1)
a + 5b = 22 (2)
Subtracting equation (2) from equation (1),
Substituting a = 2 in equation (2),
2 + 5b = 22
∴ 5b = 22 – 2
∴5b = 20
∴b = 4
Ans. (a, b) = (2, 4) is the solution.
(2) x + 7y = 10; 3x – 2y = 7
Solution:
x + 7y = 10 (1)
3x – 2y = 7 (2)
Multiplying equation (1) by 3,
3x + 21y = 30
Subtracting equation (2) from equation (3),
Substituting y = 1 in equation (1),
x + 7(1) = 10
∴x + 7 = 10 ∴x = 10 – 7 ∴ x = 3
Ans. (x, y) = (3, 1) is the solution.
(3) 2x – 3y = 9; 2x + y = 1
Solution:
2x – 3y = 9 (1)
2x + y = 1 (2)
Subtracting equation (1) from equation (2),
Substituting y = 1 in equation (2),
2x + 1 = 13
∴2x = 13 – 1 ∴ 2x=12 ∴x=6
Ans. (x, y) = (6, 1) is the solution.
(4) 5m-3n=19 ; m – 6n = – 7
Solution :
5m – 3n = 19 (1)
m – 6n = – 7 (2)
m = – 7 + 6n (3)
Substituting this value of m in equation (1),
5(- 7 + 6n) – 3n = 19
∴- 35 + 30n – 3n = 19
∴ 27n = 19 + 35
∴ 27n = 54
∴n = 2 .. (Dividing both the sides by 27)
Substituting n = 2 in equation (3),
m = – 7 + 6(2)
∴ m = – 7 + 12 ∴m =5
Ans. (m, n) = (5, 2)
(5) 5x + 2y = -3; x + 5y =4
Solution:
5x + 2y = -3 (1)
x+5y=4 (2)
∴ x=4-5y (3)
Substituting this value of x in equation (1),
5(4-5y)+2y= -3 .. 20-25y+2y= -3
∴23y=-3-20
∴23y=-23
∴y = 1 [Dividing both the sides by (-23)] Substituting y = 1 in equation (3),
x=4-5(1)
∴x=4 – 5
x= -1
Ans. (x, y) = (-1, 1) is the solution.
(7) 99x + 101y = 499; 101x + 99y = 501
99x + 101y=499(1)
101x + 99y = 501(2)
[Here, the coefficients of two variables (x and y) in one equation are interchanged in the other equation. By adding and subtracting such equations, we get, the new equations in the formx x + y = a and x – y = b These equations can be easily solved.] Adding equations (1) and (2),
(8) 49x – 57y = 172; 57x – 49y = 252
49x – 57y = 172 (1)
57x – 49y = 252 (2)
Adding equation (1) and (2)
