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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

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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:

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

Considering equation

space space space space space space space space space fraction numerator 7 straight x minus 2 straight y over denominator x y end fraction equals 5

rightwards double arrow space space space space space space 7 straight x minus 2 straight y equals 5 xy

Dividing both sides by xy, we get

fraction numerator 7 straight x over denominator xy end fraction minus fraction numerator 2 straight y over denominator xy end fraction equals fraction numerator 5 xy over denominator xy end fraction rightwards double arrow space space 7 over straight y minus 2 over straight x equals 5 space space space space space space space space space space space space space space space space... left parenthesis straight i right parenthesis

Considering equation

fraction numerator 8 straight x plus 7 straight y over denominator xy end fraction equals 15 rightwards double arrow space 8 straight x plus 7 straight y equals 15 straight y

Dividing both sides by xy, we get

space space space space fraction numerator 8 straight x over denominator xy end fraction plus fraction numerator 7 straight y over denominator xy end fraction equals fraction numerator 15 xy over denominator xy end fraction space rightwards double arrow space space 8 over straight y plus 7 over straight x equals 15 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 left parenthesis ii right parenthesis

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

Then the given system of equations become
7v - 2u = 5 ...(iii)
8v + 7u = 15 ...(iv)
For making the coefficient of ‘u’ in (iii) and (iv) equal, we multiply (iii) by ‘7’ and (iv) by ‘2’ and then adding

space space space space 49 v minus 14 u space equals space 35 bottom enclose space space space 16 v space plus 14 u space equals space 30 space space space end enclose space space space space 65 v space plus space 0 space space equals space 65 rightwards double arrow space space space space space v space equals space 1

Putting the value of V in (iii), we get

Considering equationDividing both sides by xy, we getConsidering equa

Hence, x = 1 ,y = 1 is the solution of the given equation.

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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:

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

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

158 Views

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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Previous Year Papers

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Short Answer Type

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