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Pair of Linear Equations in Two Variables

Short Answer Type

131. Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method :

Places A and B are 100 km apart on a highway. One 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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132. Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method :

The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.

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133. Solve the following pairs of equations by reducing them to a pair of linear equations :

$fraction numerator 1 over denominator 2 straight x end fraction plus fraction numerator 1 over denominator 3 straight y end fraction equals 2 fraction numerator 1 over denominator 3 straight x end fraction plus fraction numerator 1 over denominator 2 straight y end fraction equals 13 over 6$

Let space 1 over straight x equals straight u space and space 1 over straight y equals straight v

Then, the given system of equation becomes

1 half straight u plus 1 third straight v equals 2 rightwards double arrow space space space 3 straight u plus 2 straight v equals 12 and space 1 third straight u plus 1 half straight v equals 13 over 6 rightwards double arrow space fraction numerator 2 straight u plus 3 straight v over denominator 6 end fraction equals 13 over 6 rightwards double arrow space 2 straight u plus 3 straight v equals 13

Thus, we have two equations
3u + 2v = 12 ...(i)
2u + 3v = 13 ...(ii)
From (i), we have

$straight u space equals space fraction numerator 12 minus 2 straight v over denominator 3 end fraction space space space... left parenthesis iii right parenthesis$
Putting the value of ‘u’ in (ii), we get

$2 open parentheses fraction numerator 12 minus 2 v over denominator 3 end fraction close parentheses plus 3 v equals 13 rightwards double arrow space 24 minus 4 v plus 9 v equals 39 rightwards double arrow space 5 v space equals space 39 minus 24 rightwards double arrow space space 5 v equals 15 rightwards double arrow space space space v space equals space 3$

Putting the value of ‘v’ in (iii), we get

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$H e n c e comma space straight x space equals space 1 half comma space space straight y space equals 1 third space space$ is the solution of the given equation.

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134. Solve the following pairs of equations by reducing them to a pair of linear equations :

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fraction numerator 4 over denominator square root of straight x end fraction minus fraction numerator 9 over denominator square root of 9 end fraction equals negative 1

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135. Solve the following pairs of equations by reducing them to a pair of linear equations :

4 over straight x plus 3 straight y equals 14 3 over straight x minus 4 straight y equals 23

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136.

Solve the following pairs of equations by reducing them to a pair of linear equations:

$fraction numerator 5 over denominator x minus 1 end fraction plus fraction numerator 1 over denominator y minus 2 end fraction equals 2$

$fraction numerator 6 over denominator straight x minus 1 end fraction minus fraction numerator 3 over denominator straight y minus 2 end fraction equals 1$

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137.

Solve the following pairs of equations by reducing them to a pair of linear equations:

$space space space space space space space space space fraction numerator 7 straight x minus 2 straight y over denominator xy end fraction equals 5 space space space space space space space fraction numerator 8 straight x plus 7 straight x over denominator x y end fraction equals 15$

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138.

Solve the following pairs of equations by reducing them to a pair of linear equations:

6x + 3y = 6xy
2x + 4y = 5x

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139.

Solve the following pairs of equations by reducing them to a pair of linear equations:

$fraction numerator 10 over denominator straight x plus straight y end fraction plus fraction numerator 2 over denominator straight x minus straight y end fraction equals 4$

$fraction numerator 15 over denominator straight x plus straight y end fraction minus fraction numerator 5 over denominator straight x minus straight y end fraction equals negative 2$

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140.

Solve the following pairs of equations by reducing them to a pair of linear equations:

$fraction numerator 1 over denominator 3 x plus y end fraction plus fraction numerator 1 over denominator 3 x minus y end fraction space equals space 3 over 4 fraction numerator 1 over denominator 2 left parenthesis 3 x plus y right parenthesis end fraction plus fraction numerator 1 over denominator 2 left parenthesis 3 x minus y right parenthesis end fraction space equals space fraction numerator negative 1 over denominator space space 8 end fraction$

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