## Chapter 1 – Linear Equations in Two Variables

## Practice Sums

**1 Mark Questions:**

(1) What is the general form of linear equation in two variables?

(2) Find one of the solutions of equation 3x – 5y = 10.

(3) If 12x + 13y = 29 and 13x + 12y = 21 then the value of x + y is _____

(4) Express the following information in mathematical form using x and y variables:

One number is 5 more than seven times the other number.

(5) Write Dx for the following simultaneous equations:

3x + 4y = 8 ; x – 2y = 5

(6) What is the solution of given simultaneous equations?

x – y = 7, x + y = 11.

(7) When we consider two linear equations in two variables, what are such equations called?

(8) If the value of the determinant is \(\begin{vmatrix}M&2\\2&1\end{vmatrix}\) is 7, then what is the value of m?

(9) The perimeter of a rectangle is 64. How is it expressed in the mathematical equation form?

(10) What is the value of determinant \(\begin{vmatrix}5&2\\7&4\end{vmatrix}\) ?

(11) If Dx = – 18 and D = 3 are values of determinant for certain simultaneous equation in x and y, then what is the value of x?

(12) Find m if value of determinant \(\begin{vmatrix}m&2\\5&7\end{vmatrix}\) is 31.

(13) If (a, 3) is point lying on graph of equation 5x + 2y = – 4 then what is the value of a?

(14) By Cramer’s rule, what is the value of x and y?

(1) 15 cm

(2) 16 : 81

(3) 3 : 4

(4) 7.5 cm

(5) 7.5 cm

(6) 5 : 2

(7) 20.4 cm

(8) 7 : 11

(9) 15 units

(10) 10

(11) 2 : 3

(12) 7.5 cm

**2 Marks Questions:**

**Solve the following simultaneous equations:**

(i) 3x – y = 2 ; 5x – 2y = 1

(ii) 4m + 3n = 18 ; 3m – 2n = 5

(iii) 2x – 3y = 14 ; 5x + 2y = 16

(iv) x + y = 15 ; x – y = 3

(v) x + y = 0 ; x – y = 2

(vi) 2x – y = 3 ; 4x + y = 3

(vii) 2x – 9y = 9 ; 5x + 2y = 27

(viii) x + 4y = – 4 ; 3y – 5x = – 1

(ix) 2x – 3y = 2 ; x + 2y = 8

(x) x + y = 7 ; 2x – 3y = 9

(xi) 11y + 15x = – 23 ; 7y – 2x = 20

(xii) 5x – 6y = 2 ; 6x – 5y = 9

(1) 12 cm

(2) 9.6 units

(3) 20 units

(5) 42 m

(7) 25 : 36

(8) 2.8

**3 Marks Questions:**

**Solve the following simultaneous equations:**

(i) \(\large \frac {x}{6}\) + \(\large \frac {y}{15}\) = 4 ; \(\large \frac {x}{3}\) – \(\large \frac {y}{12}\) = \(\large \frac {19}{4}\)

(ii) 47x + 31y = 63 ; 31x + 47y = 15

(iii) \(\large \frac {1}{3}\)x + 5y = 13 ; 2x + \(\large \frac {1}{2}\)y = 19

(iv) \(\large \frac {1}{3}\)x + \(\large \frac {1}{4}\)y = 13 ; \(\large \frac {5}{6}\)x – \(\large \frac {1}{8}\)y = 4

(v) 64x – 45y = 289 ; 45x – 64y = 365

(i) x = 18, y = 15

(ii) x = 2, y = – 1

(iii) x = 9, y = 2

(iv) x = 6, y = 8

#### (v) x = 1, y = – 5

**4 Marks Questions:**

**(1) Solve the following simultaneous equations using Graphical method:**

(i) x + y = 8 ; x – y = 2

(ii) 3x + 4y = – 5 ; x – y = – 4

(iii) x + 3y = 7 ; 2x + y = – 1

(iv) x + 2y = 5 ; 2x + y = – 2

(v) 4x – y = – 5 ; 2x – y = – 1

**(2) Solve the following simultaneous equations using Cramer’s method:**

(i) 3x – 2y = 3; 2x + y = 16

(ii) x + 2y + 4 = 0; 3x = – 4y – 16

(iii) 3x – y = 7; x + 4y = 11

(iv) 3x + y = 1; 2x = 11y + 3

(v) 4x + 3y = 4 ; 6x + 5y = 8

**(3) Solve the following simultaneous equations:**

(i) \(\large \frac {4}{x}\) + \(\large \frac {3}{y}\) = 1 ; \(\large \frac {8}{x}\) – \(\large \frac {9}{y}\) = 7

(ii) \(\large \frac {7}{2x \,+\, 1}\) + \(\large \frac {13}{y\,+\,2}\) = 27 ; \(\large \frac {13}{2x \,+\, 1}\) + \(\large \frac {7}{y\,+\,2}\) = 33

(iii) \(\large \frac {14}{x \,+\, y}\) + \(\large \frac {3}{x\,–\,y}\) = 5 ; \(\large \frac {21}{x \,+\, y}\) – \(\large \frac {2}{x\,–\,y}\) = 1

(iv) \(\large \frac {5}{x \,–\, 1}\) + \(\large \frac {1}{y\,–\,2}\) = 2 ; \(\large \frac {6}{x \,–\, 1}\) – \(\large \frac {3}{y\,–\,2}\) = 1

(v) \(\large \frac {4}{x\,–\, 3}\) + \(\large \frac {6}{y\,–\,4}\) = 5 ; \(\large \frac {5}{x \,–\, 3}\) – \(\large \frac {3}{y\,–\,4}\) = 1

**(4) Solve the following simultaneous equations:**

(i) Shabana’s age 10 years hence, will be twice Juhi’s present age. 6 years back shabana’s age was \(\large \frac {5}{3}\) times Juhi’s at that time find their present ages.

(ii) If 1 is added to the numerator of a certain fraction its value becomes \(\large \frac {1}{2}\) and if 1 is added to its denometer \(\large \frac {1}{3}\). Find the original fraction.

(iii) Sum of two numbers is 45 and the greater number is twice the smaller number. Find the numbers.

(iv) A man travels 370 km partly by train and partly by car. If he covers 250 km by train and the rest by car, it takes him 4 hours. But, If he travels 130 km by train and the rest by car, he takes 18 minutes longer. Find the speed of the train and that of the car.

(v) The sum of present ages of Madhu and Raju is 11 years. Madhu is older than Raju by 9 years. Find their present ages.

(vi) The monthly incomes of A and B are in the ratio 8:7 and their expenditures are in the ratio 19:16. If each saves ₹ 5000 per month, find the monthly income of each.

(vii) The sum of a two-digit number obtained by reversing the order of its digits is 99. If the digits differ by 3, find the original number.

(viii) A man’s age is three times the sum of the ages of his two sons. After 5 years, his age will be twice the sum of his two son’s age. Find the age of the man.

(ix) A man can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours, Find his speed of rowing in still water. Also, find the speed of the stream.

(x) Find the four angles of a cyclic Quadrilateral ABCD in which ∠A = (2x – 1)⁰, ∠B = (y + 5)⁰, ∠C = (2y + 15)⁰, and ∠D = (4x – 7)⁰.

(xi) 8 men and 12 boys can finish a piece of work in 5 days, while 6 men and 8 boys can finish it in 7 days. Find time taken by a man and a boy alone to finish the same work.

(xii) Five years ago Amaan was three times older than Rehan and ten years later Amaan will be two times older than Rehaan. What are the present ages of Amaan and Rehan?

**(1)**

(i) x = 5, y = 3

(ii) x = – 3, y = 1

(iii) x = – 2, y = 3

(iv) x = – 3, y = 4

#### (v) x = – 2, y = – 3

**(2)**

(i) x = 5, y = 6

(ii) x = – 8, y = 2

(iii) x = 3, y = 2

(iv) x = \(\large \frac {2}{5}\), y = – \(\large \frac {1}{5}\)

#### (v) x = – 2, y = 4

**(3)**

(i) x = 2, y = – 3

(ii) x = – \(\large \frac {1}{4}\), y = – 1

(iii) x = 4, y = 3

(iv) x = 4, y = 5

#### (v) x = 5, y = 6

**(4)**

(i) 26 years, 18 years.

(ii) \(\large \frac {3}{8}\)

(iii) 30, 15.

(iv) 100 km/hr and 80 km/hr.

(v) Present age of Madhu is 10 years and of Raju is 1 year.

(vi) A = ₹24,000, B = ₹21,000

(vii) 63

(viii) 45 years

(ix) Speed of the man = 6 km/hr and speed of the current = 4 km/hr.

(x) 65⁰, 55⁰, 115⁰, 125⁰

(xi) Man = 70 days and Boy = 140 days.

(xii) Present age of Amaan is 50 years and of Rehan is 20 years.

**(1)**

(i) x = 5, y = 3

(ii) x = – 3, y = 1

(iii) x = – 2, y = 3

(iv) x = – 3, y = 4

#### (v) x = – 2, y = – 3

**(2)**

(i) x = 5, y = 6

(ii) x = – 8, y = 2

(iii) x = 3, y = 2

(iv) x = \(\large \frac {2}{5}\), y = – \(\large \frac {1}{5}\)

#### (v) x = – 2, y = 4

**(3)**

(i) x = 2, y = – 3

(ii) x = – \(\large \frac {1}{4}\), y = – 1

(iii) x = 4, y = 3

(iv) x = 4, y = 5

#### (v) x = 5, y = 6

**(4)**

(i) 26 years, 18 years.

(ii) \(\large \frac {3}{8}\)

(iii) 30, 15.

(iv) 100 km/hr and 80 km/hr.

(v) Present age of Madhu is 10 years and of Raju is 1 year.

(vi) A = ₹24,000, B = ₹21,000

(vii) 63

(viii) 45 years

(ix) Speed of the man = 6 km/hr and speed of the current = 4 km/hr.

(x) 65⁰, 55⁰, 115⁰, 125⁰

(xi) Man = 70 days and Boy = 140 days.

(xii) Present age of Amaan is 50 years and of Rehan is 20 years.