ML Aggarwal Solutions Class 9 Math 5th Chapter Simultaneous Linear Equations Exercise 5.4
ML Aggarwal Understanding ICSE Mathematics Class 9 Solutions Fifth Chapter Simultaneous Linear Equations Exercise 5.4. APC Solution Class 9 Exercise 5.4.
(1) (i) 2/x + 2/3y = 1/6
2/x – 1/y = 1
Solution:
2/x + 2/3y = 1/6
Let, 1/x = p, 1/y = m
∴ 2p + 2m/3 = 1/6
Or, 6p + 2m = 1/2 —- (i)
2/x – 1/y = 1
Or, 2p – m = 1 — (ii)
Multiplying equation (ii) by 2 then adding them we get,
4p – 2m = 2
6p + 2m = 1/2
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10p = 2 + 1/2
Or, p = 4+1/2×10
Or, p = 5/2×10
Or, p = 1/4
Substituting the value of p in equation (ii) we get,
2 × 1/4 – m = 1
Or, m = 1 – 1/2
Or, m = 1/2
Substituting the value of p & m into x and y we get,
p = 1/x
Or, 1/x = 1/4
Or, x = 4
m = 1/2
Or, 1/y = 1/2
Or, y = 2
(ii) 3/2x + 2/3y = 5
5/x – 3/y = 1
Solution:
3/2x + 2/3y = 5,
Let, 1/x = m, 1/y = n
∴ 3m/2 + 2n/3 = 5
Or, 9m+4n/6 = 5
Or, 9m + 4n = 30 — (i)
5/x – 3/y = 1
5m – 3n = 1 —- (ii)
Multiplying equation (i) with 4 & (ii) with 3 then adding then we get,
20m – 12n = 4
27m + 12n = 90
_________________
47m = 94
Or, m = 2
Substituting the value of m in equation (i) we get,
9×2 + 4n = 30
Or, 4n = 30 – 18
Or, n = 12/4
Or, n = 3
Substituting the value of m & n with 1/x & 1/y we get,
m = 2
Or, 1/x = 2
Or, x = 1/2
n = 3
Or, 1/y = 3
Or, y = 3
(2) (i) 7x-2y/xy = 5
8x+7y/xy = 15
Solution:
(i) 7x-2y/xy = 5
Or, 7x/xy – 2y/xy = 5
Or, 7/y – 2/x = 5
8x + 7y/xy = 15
Or, 8x/xy + 7y/xy = 15
Or, 8/y + 7/x = 15
Let, 1/x = m, 1/y = n
∴ 7n – 2m = 5 —- (i) 8n + 7m = 15 —– (ii)
Multiplying equation (i) with 7 & 2 with 2 then adding there we get,
49n – 14m = 35
16n + 14m = 30
___________________
55n = 55
Substituting the value of n in equation (i) we get,
7 × 1 – 2m = 5
Or, -2m = 5 – 7
Or, -m = -2/2
Or, m = 1
Substituting the values of m & n as 1/x & 1/y we get
n = 1
Or, 1/y = 1
Or, y = 1
m = 1
Or, 1/x = 1
Or, x = 1
(ii) 99x + 101y = 499xy
101x + 99y = 501xy
Solution:
99x + 101y = 449xy
Or, 99x+101y/xy = 499
Or, 99/y + 101/x = 499
Let, 1/x = m, 1/y = n
∴ 99n + 101m = 499 —- (i)
101x + 99y = 501xy
Or, 101x+99y/xy = 501
Or, 101/y + 99/x = 501
101n + 99m = 501 —- (ii)
Adding equation (i) & (ii) we get,
99n + 101m = 499
101n + 99m = 501
_________________
200n + 200m = 1000
Or, 200 (n + m) = 1000
Or, n + m = 5 —- (iii)
Subtracting equation (ii) from equation (i) we get,
101n + 99m = 501
99n + 101m = 499
(-) (-) (-)
___________________
2n – 2m = 2
Or, 2 (n – m) = 2
Or, n – m = 1 — (iv)
Adding equation (iii) & (iv) we get,
n + m = 5
n – m = 1
___________
2n = 6
Or, n = 3
Substituting the value of n in equation (iii) we get,
3 + m = 5
Or, m = 2
substituting the values of m & n as 1/x, 1/y we get,
n = 3
Or, 1/y = 3
Or, y = 1/3
m = 2
Or, 1/x = 2
Or, x = 1/2
(3) (i) 3x +14y = 5xy
21y – x = 2xy
Solution:
3x + 14y = 5xy
Or, 3x + 14y/xy = 5
Or, 3/y + 14/x = 5
Or, 14/x + 3/y = 5 —- (i)
21y – x = 2xy
Or, 21y-x/xy = 2
Or, 21/x – 1/y = 2 — (ii)
∴ 14m + 3n = 5 —- (iii)
Let, 1/x = m, 1/y = n
21m – n = 2 (iv)
Multiplying equation (iv) with 3 then adding equation (iii) & (iv) we get,
14m + 3n = 5
63m – 3n = 6
________________
77m = 11
Or, m= 1/7
Substituting the value of m in equation (i) we get,
14 × 1/7 + 3n = 5
Or, 3n = 5 – 2
Or, 3n = 3
Or, n = 1
Substituting the value of m & n as 1/y we get,
m = 1/7
Or, 1/x = 1/7
Or, x = 7
n = 1
Or, 1/y = 1
Or, y = 1
(ii) 3x + 5y = 4xy
2y – x = xy
Solution:
3x + 5y = 4xy
Or, 3x+5y/xy = 4
Or, 3/y + 5/x = 4
Or, 5/x + 3/y = 4 —- (i)
2y – 2 = xy
Or, 2y-x/xy = 1
Or, 2/x – 1/y = 1 —- (ii)
∴ 5m + 3n = 4 —- (iii)
2m – n = 1 — (iv)
Multiplying equation (iv) with 3 then adding equation (iii) & (iv) we get,
5m + 3n = 4
6m – 3n = 3
________________
11m = 7
Or, m = 7/11
Substituting the value of m in equation (iii) we get
2 × 7/11 – n = 1
Or, n = 14/11 – 1
Or, n = 3/11
Substituting the value of m & n as 1/x & 1/y we get,
m = 7/11
Or, 9 1/x = 7/11
Or, x = 11/7
n = 3/11
Or, 1/4 = 3/11
Or, y = 11/3
(4) (i) 20/x+1 + 4/y-1 = 5
10/x+1 – 4/y-1 = 1
Solution:
20/x+1 + 4/y-1 = 5
10/x+1 – 4/y-1 = 1
Let, 1/x+1 = m, 1/y-1 = n
∴ 20m + 4n = 5 —- (i)
10m – 4n = 1 —- (ii)
Adding equation (i) & (ii) we get,
20m + 4n = 5
10m – 4n = 1
________________
30m = 6
Or, m = 1/5
Substituting the value of m in equation (ii) we get,
10 × 1/8 – 4n = 1
Or, 4n = 2 – 1
Or, n = 1/4
Substituting the value of m & n as 1/x+1, 1/y-1, we get
m = 1/5
Or, 1/x+1 = 1/5
Or, x+1 = 5
Or, x = 4
n = 1/4
Or, 1/y-1 = 1/4
Or, y – 1 = 4
Or, y = 5
(ii) 3/x+y + 2/x-y = 3
2/x+y + 3/x-y = 11/3
Solution:
3/x+y + 2/x-y = 3, 2/x+y + 3/x-y = 11/3
Let, 1/x+y = m, 1/x-y = n
∴3m + 2n = 3 —- (i), 2m + 3n = 11/3 —– (ii)
Multiplying equation (i) with 2 & (ii) with 3 we get, then subtract equation (ii) from (i)
6m + 4n = 6
6m + 9n = 11
(-) (-) (-)
_____________
-5n = -5
Or, n = 1
Substituting the value of m in equation (i) we get,
3m + 2 × 1 = 3
Or m = 3-2/3
Or, m = 1/3
Putting the values of m & n
n = 1
Or, 1/x-y = 1
Or, x – y = 1 —– (iii)
m = 1/3
Or, 1/x+y = 1/3
Or, x + y = 3 — (iv)
Adding equation (iii) & (iv) we get,
x-b = 1
x+y = 3
__________
2x = 4
Or, x = 2
Substituting the value of x in equation (iv) we get,
2 + y = 3
Or, y = 1
(5) (i) 1/2(2x+3y) + 12/7(3x-2y) = 1/2
7/2x + 3y + 4/3x – 2y = 2
Solution:
1/2(2x+3y) + 12/7(3x-2y) = 1/2, 7/2x+3y + 4/3x-2y = 2
Let, 1/2x+3y = m, 1/3x-2y = n
∴ m/2 + 12n/7 = 1/2, 7m + 4n = 2 —- (ii)
Or, 7m+24n/14 = 1/2
Or, 7m + 24n = 7 —- (i)
Subtracting equation (ii) from equation (i) we get,
7m + 24n = 7
7m + 4n = 2
(-) (-) (-)
__________________
20n = 5
Or, n = 1/4
Substituting the value of m in equation (i) we get,
7m + 24 × 1/4 = 7
Or, m = 1/7
Putting the values of m & n as 1/2x+3y & 1/3x-2y we get
n = 1/4
Or, 1/3x – 24 = 4 —– (iii)
m= 1/7
Or, 1/2x + 3y = 1/7 —- (iv)
Multiplying equation (iii) with 3 & (iv) with 2 then adding these we get,
9x – 6y = 12
4x + 6y = 14
____________
13x = 26
Or, x = 2
Substituting the values of x from equation (iii) we get
3 × 2 – 2y = 4
Or, -24 = 4 – 6
Or, – 4 = -2/2
Or, y = 1
(ii) 1/2(x+2y) + 5/3(3x-2y) = -3/2
5/4(x+2y) – 3/5(3x-2y) = 61/60
Solution:
1/2(x+2y) + 5/3(3x-2y) = -3/2
Let, 1/x+2y = m, 1/3x-2y = n
∴ m/2 + 5n/3 = -3/2
Or, 3m+10n/6 = -3/2
Or, 3m + 10n = -9 — (i)
5/4(x+2y) – 3/5(3x-2y) = 61/60
5m/4 – 3n/5 = 61/60
Or, 25m-12n/20 = 61/60
Or, 25m – 12n = 61/3 —- (ii)
Multiplying equation (i) with 6 & (ii) with 5 then adding them we get,
18m + 60n = – 54
125m – 60n = 305/3
_________________________
143m = 305/3 – 54
Or, 143m = 305-162/3
Or, 143m = 143/3
Or, m = 1/3
Substituting the values of m in equation (i) we get,
3 × 1/3 + 10n = -9
Or, n = -9-1/10
Or, n = -1
Putting the values of m & n as 1/x+2y & 1/3x-2y we get,
m = 1/3
Or, 1/x+2y = 1/3
Or, x + 2y = 3 — (iii)
n = -1
Or, 1/3x-2y = -1
Or, 3x – 2y = -1 —– (iv)
Adding equation (iii) & (iv) we get,
x + xy = 3
3x – xy = -1
_____________
4x = 2
Or, x = 1/2
Substituting the value of x in equation (ii) we get,
1/2 + 24 = 3
Or, 2y = 3 – 1/2
Or, y = 5/2×2
Or, y = 5/4