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Pair of Linear Equations in Two Variables

Short Answer Type

201. Places A and B are 100 km. apart on a highway. On car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars ?

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202. The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each saves Rs. 200 per month find their monthly incomes.

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203. 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.

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204. The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?

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205.

Solve the following system of equations by elimination method

6(ax + by) = 3a + 2b
6(bx - ay) =3b - 2a.

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206. A man has only 20 paisc coins and 25 paise coins in his purse. If he has 50 coins in all, totalling Rs. 11.25. How many coins of each type does he share ? (Use Elimination Method).

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207. A person starts his job with a monthly income and earns a fixed increment every year. If his salary was Rs. 4500 after 4 years of service and Rs. 5400 after 10 years of service, find the initial salary and the annual increment by using elimination method.

Let the initial salary be ‘x’ and increment per year be ‘y’
Case I. Initial salary = x
Increment after 4 years = 4y
According to question
x + 4y = 4500
Case II. Initial salary = x
Increment after 10 years = 10y
According to question
x + 10y = 5400
Thus, we have x + 4y = 4500 ...(i)
x + 10y = 5400 ...(ii)
Since the coefficient of ‘x’ in (i) and (ii) are equal.
So we can simply eliminate the variable ‘x’ by subtracting.

$bottom enclose x space plus space 4 y space equals space 4500 x space plus space 10 y space equals space 5400 minus space minus space space space space space space space space space space minus space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space end enclose space space space space space space space minus 6 y space equals space minus 900 space rightwards double arrow space space space space space space space y space equals space 150 space$

Putting this value in (i), we get
x + 4y = 4500
⇒ x + 4(150) = 4500
⇒ x + 600 = 4500
⇒ x = 3900
Hence, initial salary = Rs. 3900 and annual Increment = Rs. 150.
Problems Based on Cross-Multiplication Method

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208.

A motor boat whose speed is 24 km/h in still water takes 1 hour more to go 32km upstream than to return downstream to the same spot. Find the speed of the stream.

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Long Answer Type

209. A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?

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210.

If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k – 1, 5k) are collinear, then find the value of k.