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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

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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 :

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

Let space space space space space space space 1 over straight x equals straight a

Thus, the given equations become
4a + 3y = 14 ...(i)
3a - 4y = 23 ...(ii)
For making the coefficient of ‘y’ in (i) and (ii), we multiply (i) by 4 and (ii) by 3 and then adding, we get

space space space 16 straight a plus 12 straight y equals 56 bottom enclose space space space 9 straight a minus 12 straight y space equals 69 space space space space space end enclose 25 straight a space space space space space space space space space space equals space 125 rightwards double arrow space space space space space space straight a space equals space 125 over 25 rightwards double arrow space space space space space space straight a equals 5

Putting the value of ‘a’ in (i), we get
4a + 3y = 14

rightwards double arrow space space space space space space space 4 left parenthesis 5 right parenthesis space plus space 3 straight y space equals space 14 rightwards double arrow space space space space space space space space 20 straight y space plus space 3 straight y space equals space 14 rightwards double arrow space space space space space space space space 3 straight y space equals space minus 6 rightwards double arrow space space space space space space space space space straight y space equals space minus 2 Now space space space space space space space straight a space equals space 5 rightwards double arrow space space space space space space space space space space space 1 over straight x equals 5 rightwards double arrow space space space space space space space space space space space space straight x space equals 1 fifth

Hence,

space space space space space space space space space space space space space space straight x equals 1 fifth space and space straight y space equals negative 2

, is the solution of the given equation.

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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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