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Pair of Linear Equations in Two Variables

Long Answer Type

191.

Draw the graph of the following equation x + y = 5, 3x - y = 3. Shade the region bounded by these lines and x-axis.

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192.

Nitish went to winter sale at Kamla Nagar to purchase some shirts and jackcts. When his friend asked him how many of each he had bought, he answered. “The number of jackets is three less than four times the number of shirts purchased. Also the number of jackets is four less than five times the numbers of shirts purchased.” Help his friends to find how many shirts and jackets Nitish bought. Express the solution graphically.

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193. Solve graphically the following system of linear equation, 2x + y = 6, x - 2y = -2. Also find the co-ordinates of the points where the lines meet the A-axis.

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194.

Represent the following pair of equations graphically and write the coordinates of points where the lines intersect y-axis :
x + 3y = 6
2x - 3y = 12

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Short Answer Type

195. Solve the following system of equation by substitution method :

fraction numerator 3 straight x over denominator 2 end fraction minus fraction numerator 5 straight y over denominator 3 end fraction equals negative 2 straight x over 3 plus straight y over 2 equals 13 over 6

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196. Solve the following system of equation by substitution method :

square root of 2 straight x end root plus square root of 3 straight y end root equals 9 space space space space space space space space space space space square root of 3 straight x end root minus square root of 8 straight y end root equals 0 space space space space space space space space space space space

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197. A part of monthly expenses of a family is constant and the remaining varies with the price of wheat. When the rate of wheat is Rs. 250 a quintal, the total monthly expenses of the family are Rs. 1000 and when it is 240 a quintal, the total monthly expenses are Rs. 980. Find the total monthly expenses of the family when the cost of wheat is Rs. 350 a quintal.

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198. A two digit number is obtained by either multiplying the sum of digits by 8 and adding 1 or by multiplying the difference of the digits by 13 and adding 2. Find the number.

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199. A and B are friends and their ages differ by 2 years. A’s father D is twice as old as A and B is hvicc as old as his sister C. The age of D and C differ by 40 years. Find the ages of A and B.

Let the ages of A and B be x and y years respectively. Then,

space space space space space space straight x minus straight y minus plus-or-minus 2 space space space space space space space left square bracket Given right square bracket

D’s age = 2x years and C’s age

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Clearly, D is older than C

therefore space space space space space 2 straight x minus straight y over 2 equals 40 rightwards double arrow 4 straight x minus straight y equals 80

Thus, we have the following systems of linear equations
x - y = 2    ...(i)
4x - y = 80    ...(ii)
and    x - y = -2    ...(iii)
Case I : From (i), we have
[Considering (i) and (ii)]
x - y = 2
⇒    x = 2 + y    ...(iv)
Substituting the value of x in (ii), we get
4x - y - 80
⇒ 4(2 + y) - y = 80
⇒ 8 + 4y - y = 80
⇒ 8 + 3y = 80 ⇒ 3y = 72
⇒    y = 24
Substituting the value ofy in (iv), we get
x = 2 + y = 2 + 24=    26
Hence, Age of A = 26 years
and Age of B = 24 years
Case II : From (iii), we have
[Considering (ii) and (iii)]
x - y = -2 ⇒ x = y - 2 ...(v)
Putting the value of (v) in (ii), we get
4x - y = 80

$rightwards double arrow 4 left parenthesis straight y minus 2 right parenthesis minus straight y space equals space 80 rightwards double arrow 4 straight y minus 8 minus straight y equals 80 rightwards double arrow space 3 straight y space equals space 88 rightwards double arrow space space space space straight y space equals space 88 over 3 equals 29 1 third$
Substituting the value ofy in (v), we get

$straight x equals 88 over 3 minus 2 equals fraction numerator 88 minus 6 over denominator 3 end fraction equals 82 over 3 equals 27 1 third Hence comma space Age space of space straight A space equals space 29 1 third space space years Age space of space straight B space equals space 27 1 third space years$

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200. A man sold a chair and a table together for Rs. 1520 thereby making a profit of 25% on the 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.

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