Solve the following pair of linear equations by the substitution method.
Solve the following pair of linear equations by the substitution method.
(i) x + y = 14 ; x – y = 4
(ii) s – t = 3 ; s/3 + t/2 = 6
(iii) 3x – y = 3 ; 9x – 3y = 9
(iv) 0.2x + 0.3y = 1.3 ; 0.4x + 0.5y = 2.3
(v) √2x+ √3y = 0 ; √3x - √8y = 0
(vi) 3/2x - 5/3y = -2 ; x/3 + y/2 = 13/6
Answer
(i) x + y = 14 ... (i)
x – y = 4 ... (ii)
From equation (i), we get
x = 14 - y ... (iii)
Putting this value in equation (ii), we get
(14 - y) - y = 4
14 - 2y = 4
10 = 2y
y = 5 ... (iv)
Putting this in equation (iii), we get
x = 9
∴ x = 9 and y = 5
(ii) s – t = 3 ... (i)
s/3 + t/2 = 6 ... (ii)
From equation (i), we gets = t + 3
Putting this value in equation (ii), we get
t+3/3 + t/2 = 6
2t + 6 + 3t = 36
5t = 30
t = 30/5 ... (iv)
Putting in equation (iii), we obtain
s = 9
∴ s = 9, t = 6
(iii) 3x - y = 3 ... (i)
9x - 3y = 9 ... (ii)
From equation (i), we get
y = 3x - 3 ... (iii)
Putting this value in equation (ii), we get
9x - 3(3x - 3) = 9
9x - 9x + 9 = 9
9 = 9
This is always true.
Hence, the given pair of equations has infinite possible solutions and the relation between these variables can be given by
y = 3x - 3
Therefore, one of its possible solutions is x = 1, y = 0.
(iv) 0.2x + 0.3y = 1.3 ... (i)
0.4x + 0.5y = 2.3 ... (ii)
0.2x + 0.3y = 1.3
Solving equation (i), we get
0.2x = 1.3 – 0.3y
Dividing by 0.2, we get
x = 1.3/0.2 - 0.3/0.2
x = 6.5 – 1.5 y …(iii)
Putting the value in equation (ii), we get
0.4x + 0.5y = 2.3
(6.5 – 1.5y) × 0.4x + 0.5y = 2.3
2.6 – 0.6y + 0.5y = 2.3
-0.1y = 2.3 – 2.6
y = -0.3/-0.1
y = 3
Putting this value in equation (iii) we get
x = 6.5 – 1.5 y
x = 6.5 – 1.5(3)
x = 6.5 - 4.5
x = 2
∴ x = 2 and y = 3
