LINEAR EQUATIONS IN TWO VARIABLES

EX. NO. 3.1


1. Solve the following simultaneous equations using graphical method :


(i) x + y = 8, x – y = 2 [Ans.]


(ii) 3x + 4y + 5 = 0; y = x + 4 [Ans.]


(iii) 4x = y – 5; y = 2x + 1 [Ans.]


(iv) x + 2y = 5; y = – 2x – 2 [Ans.]




(v) 2x + y = 6;   4 – 3x/4 = y [Ans.]





EX. NO. 3.2



1. Find the value of the following determinants:



i.
5
2

7
4


ii.
-3
8

6
0



iii.
1.2
0.03

0.57
-0.23



iv.
3√6
-4√2

5√3
2



v.
-4/7
-6/35

5
-2/5




2. Solve the following simultaneous equations using Cramer’s rule :

(i) 3x – y = 7; x + 4y = 11 [Ans.]



(ii) 4x + 3y – 4 = 0; 6x = 8 – 5y [Ans.]



(iii). y = (5x – 10)/2 ; 4x + 5 = - y [Ans.]


(iv) 3x + 2y + 11 = 0; 7x – 4y = 9  [Ans.]


(v) x + 18 = 2y; y = 2x – 9  [Ans.]




(vi) 3x + y = 1; 2x = 11y + 3   [Ans.]




EX. NO. 3.3


1. Without actually solving the simultaneous equations given below, decide which simultaneous equations have unique solution, no solution or infinitely many solutions.


(i) 3x + 5y = 16; 4x – y = 6   [Ans.]

(ii) 3y = 2 – x; 3x = 6 – 9y   [Ans.]

(iii) 3x – 7y = 15; 6x = 14y + 10    [Ans.]

(iv) 8y = x – 10; 2x = 3y + 7    [Ans.]

(v) (x – 2y)/3 = 1; 2x – 4y = 9/2    [Ans.]

(vi) x/2 + y/3 = 4; x/4 + y/6 = 2    [Ans.]


2. Find the value of k for which the given simultaneous equations have infinitely many solutions :

(i) 4x + y = 7; 16x + ky = 28    [Ans.]

(ii) 4y = kx – 10; 3x = 2y + 5    [Ans.]


3. Find the value of k for which are given simultaneous equations have infinitely many solutions :

(i) kx + y = k – 2; 9x + ky = k    [Ans.]

(ii) kx – y + 3 – k = 0; 4x – ky + k = 0    [Ans.]


4. Find the value of p for which the given simultaneous equations have unique solution :

(i) 3x + y = 10; 9x + py = 23     [Ans.]



(ii) 8x – py + 7 = 0; 4x – 2y + 3 = 0      [Ans.]



EX. NO. 3.4

Q. Solve the following simultaneous equations.








Ex. No. 3.5
Q. Solve the following problems using two variables.

1. The sum of two numbers is 60. The greater number is 8 more than thrice the smaller number. Find the numbers. [Ans.]

2. The perimeter of an isosceles triangle is 24 cm. The length of its congruent sides is 13 cm less than twice the length of its base. Find the lengths of all sides of the triangle.[Ans.]

3. In a right angled triangle, one of the acute angle exceeds the other by 20º. Find the measure of both the acute angles in the right angled triangle. [Ans.]

4. A house has rectangular yard in front of it for children to play. The length of that rectangle exceeds its width by 6 m and its perimeter is 60 m, find the measurement of the yard. [Ans.]

5. Seg AB is the diameter of a circle. C is the point on the circumference such that in ∆ABC, ∠B is the less by 10º than ∠A. Find the measures of all the angles of ∆ ABC.[Ans.]

(6) Durga’s mother gave some 10 rupee notes and some 5 rupee notes to her, which amounts to Rs. 190. Durga said, ‘if the number of 10 rupee notes and 5 rupee notes would have been interchanged, I would have Rs. 185 in my hand.’ So how many notes of rupee 10 and rupee 5 were given to Durga?  [Ans.]

7. A man starts his job with a certain monthly salary and a fixed increment every year. If his salary will be Rs. 11000 after 2 years and Rs. 14000 after 4 years of his service. What is his starting salary and what is the annual increment ? [Ans.]

8. AB is a segment. The point P is on the perpendicular bisector of segment AB such that length of AP exceeds length of AB by 7 cm. If the perimeter of ∆ABP is 38 cm. Find the sides of ∆ABP. [Ans.]

9. Sum of two numbers is 97. If the larger number is divided by the smaller, the quotient is 7 and the remainder is 1. Find the numbers. [Ans]

10. A boat takes 6 hours to travel 8 km upstream and 32 km downstream, and it takes 7 hours to travel 20 km upstream and 16 km downstream. Find the speed of the boat in still water and the speed of the stream. [Ans]


Problem Set No. 3. [Word Problems]











26. A bus covers a certain distance with uniform speed. If the speed of the bus would have been increased by 15 km/h, it would have taken two hours less to cover the same distance and if the speed of the bus would have been decreased by 5 km/h, it would have taken one hour more to cover the same distance. Find the distance covered by the bus.