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\(8x + 5y = 9\)

\(3x + 2y = 4\)

Answer :

Ans.(x + y = 14\)...........(i)

=> \(x – y = 4\)..............(ii)

Using equation (ii)

=> \(x = 4 + y\)......................(iii)

On substituting (iii) in (i):

=>\(4 + 2y = 14\)

=>\(2y = 10\)

=>\(y = 5\)...........(iv)

On substituting (iv) in (i):

=>\(x + 5 = 14\)

=>\(x = 14 - 5\)

=>\(x = 9\)

So, the value of \(y = 5\) and \(x = 9\).

\(x - 3y – 7 = 0,3x - 3y – 15 = 0\)

On comparing quations with general form of equation \(ax^2 + bx + c\)

\(a_1 = 1, \)

\( b_1 = -3, \)

\( c_1 = -7, \)

\( a_2 = 3, \)

\(b_2 = -3,\)

\(c_2 = -15 \)

Here \({{a_1}\over {a_2}} \ne {{b_1}\over {b_2}} \)

=>\({{1}\over {3}} \ne {{-3}\over {-3}} \)

So, these lines have a unique solution.

We can find the unique solution using cross multiplication:

=>\({{x}\over {(-3)(-15)-(-3)(-7)}} = {{y}\over {(-7)(3)-(-15)(1)}} = {{1}\over {(-3)(1)-(-3)(3)}}\)

=>\({{x}\over {45 - 21}} = {{y}\over {-21 + 15}} = {{1}\over {-3 + 9}}\)

=>\({{x}\over {24}} = {{y}\over {-6}} = {{1}\over {6}}\)

=> x = 4 and y = -1.

- Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions? In case there is a unique solution, find it by using cross multiplication method. (i) \(x – 3y – 3 = 0,3x – 9y – 2 = 0\)(ii) \(2x + y = 5,3x + 2y = 8\)(ii) \(3x - 5y = 20,6x - 10y = 40\)(ii) \(x - 3y – 7 = 0,3x - 3y – 15 = 0\)
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- Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method: (i) A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charges and the cost of food per day. (ii)A fraction becomes \({{1}\over {3}}\) when 1 is subtracted from the numerator and it becomes \({{1}\over {4}}\) when 8 is added to its denominator. Find the fraction. (iii)Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test? (iv)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? (v)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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