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Question -

Form the pair of linear equations for the following problems and find their solution by substitution method.



Answer -

(i) The difference between two numbers is 26 and one number is three times the other. Find them.
Solution:
Let the two numbers be x and y respectively, such that y > x.
According to the question,
y = 3x тАжтАжтАжтАжтАжтАж (1)
y тАУ x = 26 тАжтАжтАжтАж..(2)
Substituting the value of (1) into (2), we get
3x тАУ x = 26
x = 13 тАжтАжтАжтАжтАж. (3)
Substituting (3) in (1), we get y = 39
Hence, the numbers are 13 and 39.

(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
Solution:
Let the larger angle by xo and smaller angle be yo.
We know that the sum of two supplementary pair of angles is always 180o.
According to the question,
x + y = 180oтАжтАжтАжтАжтАж. (1)
x тАУ y = 18o тАжтАжтАжтАжтАж..(2)
From (1), we get x = 180o тАУ y тАжтАжтАжтАж. (3)
Substituting (3) in (2), we get
180o тАУ y тАУ y =18o
162o = 2y
y = 81o тАжтАжтАжтАж.. (4)
Using the value of y in (3), we get
x = 180o тАУ 81o
= 99o
Hence, the angles are 99o and 81o.

(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs.3800. Later, she buys 3 bats and 5 balls for Rs.1750. Find the cost of each bat and each ball.
Solution:
Let the cost a bat be x and cost of a ball be y.
According to the question,
7x + 6y = 3800 тАжтАжтАжтАжтАжтАж. (I)
3x + 5y = 1750 тАжтАжтАжтАжтАжтАж. (II)
From (I), we get
y = (3800-7x)/6тАжтАжтАжтАжтАжтАж..(III)
Substituting (III) in (II). we get,
3x+5(3800-7x)/6 =1750
тЗТ3x+ 9500/3 тАУ 35x/6 = 1750
тЗТ3x- 35x/6 = 1750 тАУ 9500/3
тЗТ(18x-35x)/6 = (5250 тАУ 9500)/3
тЗТ-17x/6 = -4250/3
тЗТ-17x = -8500
x = 500 тАжтАжтАжтАжтАжтАжтАжтАжтАж.. (IV)
Substituting the value of x in (III), we get
y = (3800-7 ├Ч500)/6 = 300/6 = 50
Hence, the cost of a bat is Rs 500 and cost of a ball is Rs 50.

(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs 105 and for a journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?
Solution:
Let the fixed charge be Rs x and per km charge be Rs y.
According to the question,
x + 10y = 105 тАжтАжтАжтАжтАж.. (1)
x + 15y = 155 тАжтАжтАжтАжтАж.. (2)
From (1), we get x = 105 тАУ 10y тАжтАжтАжтАжтАжтАж. (3)
Substituting the value of x in (2), we get
105 тАУ 10y + 15y = 155
5y = 50
y = 10 тАжтАжтАжтАжтАж.. (4)
Putting the value of y in (3), we get
x = 105 тАУ 10 ├Ч 10 = 5
Hence, fixed charge is Rs 5 and per km charge = Rs 10
Charge for 25 km = x + 25y = 5 + 250 = Rs 255

(v) A fraction becomes 9/11 , if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction.
Solution:
Let the fraction be x/y.
According to the question,
(x+2) /(y+2) = 9/11
11x + 22 = 9y + 18
11x тАУ 9y = -4 тАжтАжтАжтАжтАж.. (1)
(x+3) /(y+3) = 5/6
6x + 18 = 5y +15
6x тАУ 5y = -3 тАжтАжтАжтАжтАжтАж. (2)
From (1), we get x = (-4+9y)/11 тАжтАжтАжтАжтАж.. (3)
Substituting the value of x in (2), we get
6(-4+9y)/11 -5y = -3
-24 + 54y тАУ 55y = -33
-y = -9
y = 9 тАжтАжтАжтАжтАжтАжтАж (4)
Substituting the value of y in (3), we get
x = (-4+9├Ч9 )/11 = 7
Hence the fraction is 7/9.

(vi) Five years hence, the age of Jacob will be three times that of his son. Five years ago, JacobтАЩs age was seven times that of his son. What are their present ages?
Solutions:
Let the age of Jacob and his son be x and y respectively.
According to the question,
(x + 5) = 3(y + 5)
x тАУ 3y = 10 тАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАж.. (1)
(x тАУ 5) = 7(y тАУ 5)
x тАУ 7y = -30 тАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАж. (2)
From (1), we get x = 3y + 10 тАжтАжтАжтАжтАжтАжтАжтАж. (3)
Substituting the value of x in (2), we get
3y + 10 тАУ 7y = -30
-4y = -40
y = 10 тАжтАжтАжтАжтАжтАжтАж (4)
Substituting the value of y in (3), we get
x = 3 x 10 + 10 = 40
Hence, the present age of JacobтАЩs and his son is 40 years and 10 years respectively.

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