1. Linear Equations in Two Variables
1. Express y in terms of x in the equation 2x + 3y = 11. Find the point where the line represented by the equation 2x + 3y = 11 cuts yaxis.
2. Solve for x and y : + = , + = 0 and hence find ‘a’ for which y = ax – 4.
3. Solve 4x + = 15 and 6x – = 14 and hence find ‘p’ if y = px – 2.
4. For what value of k will the system of linear equations have infinite number of solutions :
kx + 4y = k – 4, 16x + ky = k ?
5. Determine the value of k for which the following system of linear equations has infinite number of solutions : (k – 3)x + 3y = k; kx + ky = 12
6. For what value of k, will the following system of equations have infinite solutions :
2x + 3y = 4, (k + 2)x + 6y = 3k + 2 ?
7. For what value of k, will the following system of equations have infinite solutions :
2x – 3y = 7, (k + 2)x – (2k + 1)y = 3(2k – 1) ?
8. A person invested some amount at the rate of 10% simple interest and some other amount at the rate of 12% simple interest. He received yearly interest of Rs. 130. But if he had interchanged the amount invested, he would have received Rs. 4 more as interest. How much amount did he invest at different rates?
9. Solve the system of equations graphically : 2(x – 1) = y; x + 3y = 15. Also, find the coordinates of the points where the lines meet the axis of y.
10. Solve graphically the system of linear equations : 2x + 3y = 12; 2y – 1 = x. Also find the coordinates of the points where the lines meet the yaxis.
11. Ramesh travels 760 km to his home partly by train and partly by car. He takes 8 hrs, if he travels 160 km by train and the rest by car. He takes 12 minutes more if he travels 240 km by train and the rest by car. Find the speed of the train and the car separately.
12. Solve for x, y and z : x + y + z = 9; 2y – z = 2; z – x = 2.
13. Solve for x and y : 631x + 279y = 910; 279x + 631y = 910.
14. Solve for x and y : 6x + 3y = 8x + 9y – 5 = 10x + 12y – 8.
15. Solve the following equation by using the method of cross multiplication :
+ = a + b; + = 2.
16. Solve the following equation by using the method of cross multiplication :
a(x + y) + b(x – y) = a2 – ab + b2
a(x + y) – b(x – y) = a2 + ab + b2
17. Ratio between the girls and boys in a class of 40 students is 2 : 3. Five new students joined the class. How many of them must be boys so that the ratio between girls and boys becomes 4 : 5 ?
18. If you travel by an autorickshaw the fare for the first kilometre is different from the rate per km for the remaining distance. The total fare for a distance of 20 km is Rs. 37.70 and that for a distance of 26 km is Rs. 48.50. Find the auto fare for the first kilometre and for each successive kilometre.
19. Ten years ago, the sum of the ages of two sons was one third of their father’s age. One son is two years older than the other and sum of their present ages is 14 years less than the father’s present age. Find the present ages of all.
20 A villager went to a hotel in a town with his big family. They consumed 23 idlies, 18 pooris, 7 dosas and 19 vadas. The bill come to Rs. 108. On next day, they consumed 34 idlies, 8 vadas, 22 pooris and 7 dosas. The bill came to Rs. 114. If an idli costs the same as a vada, what is the cost of one poori?
21. A boat goes 24 km upstream and 28 km downstream in 6 hrs. It goes 30 km upstream and 21 km downstream in 6½ hrs. Find the speed of the boat in still water and also speed of stream.
22. A dealer sold a TV and VCR for Rs. 25,820 making a profit of 15% on TV and 10% on VCR. By selling them for Rs. 25,930, he would have realised a profit of 10% on TV and 15% on VCR. Find cost price of each.
23. A man sold a chair and a table together for Rs. 1520 there by making a profit of 25% on chair and 10% on table. By selling them together for Rs. 1535 he would have made a profit of 10% on the chair and 25% on the table. Find the cost price of each.
24 Solve for x and y :
– = – ; + = 1 – x –
26. Solve for a and b : 2a + 3b = 17 and 2a + 2 – 3b + 1 = 5.
27. For what value of k, will the following system of equations have infinite solutions : 2x – 3y = 7, (k + x – (2k + 1)y = 3(2k – 1)?
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