Pair of Linear Equations in Two Variables

Chapter 3 : Pair of Linear Equations in Two Variables

Q. The coach of a cricket team buys

3

bats and

6

balls for

Rs

3900.

Later, she buys another bat and

3

more balls of the same kind for

Rs

1300.

Represent this situation algebraically.

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Q. The cost of

2 k g

of apples and

1 k g

of grapes on a day was found to be

R s 160

. After a month, the cost of

4 k g

of apples and

2 k g

of grapes is

R s 300

. Represent the situation algebraically and geometrically.

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Q. Aftab tells his daughter, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be." (Isnt this interesting?) Represent this situation algebraically and graphically.

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Q. Given the linear equation

2 x + 3 y - 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

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Q. On comparing the ratios

\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}

and

\frac{c_{1}}{c_{2}}

, find out whether the following pair of linear equations are consistent, or inconsistent.
(i)

3 x + 2 y = 5; 2 x - 3 y = 7

(ii)

2 x - 3 y = 8; 4 x - 6 y = 9

(iii)

\frac{3}{2} x + \frac{5}{3} y = 7; 9 x - 10 y = 14

(iv)

5 x - 3 y = 11; - 10 x + 6 y = - 22

(v)

\frac{4}{3} x + 2 y = 8; 2 x + 3 y = 12

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Q. Draw the graphs of the equations

x - y + 1 = 0

and

3 x + 2 y - 12 = 0

. Determine the coordinates of the vertices of the triangle formed by these lines and the

x

-axis, and shade the triangular region.

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Q. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

(i)

x + y = 5, 2 x + 2 y = 10

(ii)

x - y = 8, 3 x - 3 y = 16

(iii)

2 x + y - 6 = 0, 4 x - 2 y - 4 = 0

(iv)

2 x - 2 y - 2 = 0, 4 x - 4 y - 5 = 0

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Q. Form the pair of linear equations in the following problems, and find their solutions graphically.
(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.
(ii)

5

pencils and

7

pens together cost Rs

50,

whereas

7

pencils and

5

pens together cost Rs

46.

Find the cost of one pencil and that of one pen.

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On comparing the ratios $\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}$ , and $\frac{c_{1}}{c_{2}}$ , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

(i)

5 x - 4 y + 8 = 0; 7 x + 6 y - 9 = 0

(ii)

9 x + 3 y + 12 = 0; 18 x + 6 y + 24 = 0

(iii)

6 x - 3 y + 10 = 0; 2 x - y + 9 = 0

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Q. Why, in your opinion, did the man set the doves free?

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Q. Solve the following pair of linear equations by the substitution method.
(i)

x + y = 14; x - y = 4

(ii)

s - t = 3; \frac{s}{3} + \frac{t}{2} = 6

(iii)

3 x - y = 3; 9 x - 3 y = 9

(iv)

0.2 x + 0.3 y = 1.3; 0.4 x + 0.5 y = 2.3

(v)

\sqrt{2} x + \sqrt{3} y = 0; \sqrt{3} x - \sqrt{8} y = 0

(vi)

\frac{3 x}{2} - \frac{5 y}{3} = - 2; \frac{x}{3} + \frac{y}{2} = \frac{13}{6}

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Q. Solve

2 x + 3 y = 11

and

2 x - 4 y = - 24

and hence find the value of m for which

y = m x + 3

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Q. Form the pair of linear equations for the following problems and find their solution by substitution method.
The larger of two supplementary angles exceeds the smaller by

18

degrees. Find them.

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Solve the following pair of linear equations by the elimination method and the substitution method :

(i) x+y=5 and 2x−3y=4
(ii) 3x+4y=10 and 2x−2y=2
(iii) 3x−5y−4=0 and 9x=2y+7
(iv) x2+2y3=−1 and x−y3=3

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Q. Form the pair of linear equations in the following problems, and find their solutions(if they exist) by the elimination method :

(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes

\frac{1}{2}

if we only add 1 to the denominator. What is the fraction?

(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

(iv) Meena went to a bank to withdraw Rs.

2000

. She asked the cashier to give her Rs.

50

and Rs.

100

notes only. Meena got 25 notes in all. Find how many notes of Rs.

50

and Rs.

100

she received.

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs.

27

for a book kept for seven days, while Susy paid Rs.

21

for the book she kept for five days. Find the fixed charge and the charge for each extra day.

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Q. Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions.
(i)

x - 3 y - 3 = 0; 3 x - 9 y - 2 = 0

(ii)

2 x + y = 5; 3 x + 2 y = 8

(iii)

3 x - 5 y = 20; 6 x - 10 y = 40

(iv)

x - 3 y - 7 = 0; 3 x - 3 y - 15 = 0

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Q. Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method :

(i) A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for

20

days she has to pay

Rs. 1000

as hostel charges whereas a student B, who takes food for

26

days, pays

Rs. 1180

as hostel charges. Find the fixed charges and the cost of food per day.

(ii) A fraction becomes

\frac{1}{3}

when

1

is subtracted from the numerator and it becomes

\frac{1}{4}

when

8

is added to its denominator. Find the fraction.

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Q. (i) For which values of a and b does the following pair of linear equations have an infinite number of solutions?

2 x + 3 y = 7

;

(a - b) x + (a + b) y = 3 a + b - 2

(ii) For which value of k will the following pair of linear equations have no solution?

3 x + y = 1

;

(2 k - 1) x + (k - 1) y = 2 k + 1

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Q. Solve the following pair of linear equations by the substitution and cross-multiplication methods :

8 x + 5 y = 9

;

3 x + 2 y = 4

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Q. Solve the following pairs of equations by reducing them to a pair of linear equations:

(i)

\frac{1}{2 x} + \frac{1}{3 y} = 2; \frac{1}{3 x} + \frac{1}{2 y} = \frac{13}{6}

(ii)

\frac{2}{\sqrt{x}} + \frac{3}{\sqrt{y}} = 2; \frac{4}{\sqrt{x}} - \frac{9}{\sqrt{y}} = - 1

(iii)

\frac{4}{x} + 3 y = 14; \frac{3}{x} - 4 y = 23

(iv)

\frac{5}{x - 1} + \frac{1}{y - 2} = 2; \frac{6}{x - 1} - \frac{3}{y - 2} = 1

(v)

\frac{7 x - 2 y}{x y} = 5; \frac{8 x + 7 y}{x y} = 15

(vi)

6 x + 3 y = 6 x y; 2 x + 4 y = 5 x y

(vii)

\frac{10}{x + y} + \frac{2}{x - y} = 4; \frac{15}{x + y} - \frac{5}{x - y} = - 2

(viii)

\frac{1}{3 x + y} + \frac{1}{3 x - y} = \frac{3}{4}; \frac{1}{2 (3 x + y)} - \frac{1}{2 (3 x - y)} = \frac{- 1}{8}

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