Exercise 3.1
Form the pair of linear equations in the following problems, and find their solutions.
(i)
(i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
Solution
Given that,
Total number of students who took part in the the quiz =
Let, the number of boys who took part in the quiz = x
and, number of girls who took part in the quiz = y
According to question,
x + y =
y = x +
By putting value of x = 10 − y in equation(2), we get
y = 10 -
or, 2y =
So, y =
and x = 10 − 7 =
Therefore, Number of boys who took part in the quiz = 3
Number of girls who took part in the class = 7
(ii)
(ii)5 pencils and 7 pens together cost ₹50, whereas 7 pencils and 5 pens together cost ₹46. Find the cost of one pencil and that of one pen.
Solution
Let the cost of one pencil be Rs. x and the cost of one pen be Rs. y
According to the first condition
5x + 7y =
⇒ x =
According to the second condition
7x + 5y =
Put the value of x in eq(2)
⇒
⇒ 350 − 49y + 25y = 230 ⇒ y =
Resubstitute the value of y to find the value of x
⇒ x=
⇒ x =
∴ cost of one pencil = Rs. 3 and cost of one pen = Rs. 5
On comparing the ratios
(i)
(i) 5x - 4y + 8 = 0 and 7x + 6 - 9 = 0
Solution
a₁ =
a₂ =
From (1) and (2)
Therefore, they are
(ii)
(ii) 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0
Solution
From (1), (2) and (3)
Therefore, they are
(iii)
(iii) 6x – 3y + 10 = 0 and 2x – y + 9 = 0
Solution
a₁ =
a₂ =
From (1), (2) and (3)
Therefore, they are
On comparing the ratios
(i)
(i) 3x + 2y = 5; 2x - 3y = 7
Solution
3x + 2y = 5; 2x - 3y = 7
From the above,
Therefore, lines are
Hence, the pair of equations is
(ii)
(ii) 2x - 3y = 8; 4x - 6y = 9
Solution
2x - 3y =
From the above,
Therefore, these lines are
Hence, the pair of equations is
(iii)
(iii)
Solution
From the above,
Therefore, lines are
Hence, they are
(iv)
(iv) 5x - 3y = 11; -10x + 6y = -22
Solution
5x - 3y =
From the above,
Therefore, lines are
Hence, they are
(v)
(v)4/3x + 2y = 8; 2x + 3y = 12
Solution
From the above,
Therefore, lines are
Hence, they are
Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
Solution
Perimeter of a rectangle = 2(
Let the width(w) of the garden = x meter
Then length(l) =
Half perimeter =
So perimeter of garden =(2 × 36) =
According to the question
⇒ 2(l+b) =
⇒ 2(x + x +
⇒ 2x +
⇒ 4x =
⇒ x =
Hence,the width(w) of the garden = 16 meters
The length(l) of the garden = (16 + 4) =
Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is:
(i) intersecting lines
(ii) parallel lines
(iii) coincident lines
(i)
(i) Intersecting lines
Solution: For intersecting line, the linear equations should meet following condition:
For getting another equation to meet this criterion, multiply the coefficient of x with any number and multiply the coefficient of y with any other number. A possible equation can be as follows:
4x + 9y −
(ii)
(ii)Parallel lines
Solution: For parallel lines, the linear equations should meet following condition:
For getting another equation to meet this criterion, multiply the coefficients of x and y with the same number and multiply the constant term with any other number. A possible equation can be as follows:
4x + 6y −
(iii)
(iii) Coincident lines
Solution: For getting coincident lines, the equations should meet following condition;
For getting another equation to meet this criterion, multiply the whole equation with any number. A possible equation can be as follows:
4x + 6y −