Linear Equation in Two Variables Questions

ABCD is a cyclic quadrilateral as shown in figure. Find the angles of the cyclic quadrilateral.

Solution : We know that the sum of opposite angles of cyclic quadrilateral is 180 degrees. Angles A and C, Angles B and D form pairs of opposite angles in the given cyclic quadrilateral ABCD. $$\angle$$A + $$\angle$$C = 180   and  $$\angle$$B + $$\angle$$D = 180 $$\implies$$  (4y + 20) + 4x = 180   and  …

Solve the following pair of linear equations

Question : Solve the following pair of linear equations : (i)   px + qy = p – q       and    qx – py = p + q (ii)  ax + by = c     and    bx + ay = 1 + c (iii)  $$x\over a$$ – $$y\over b$$ = 0      …

Draw the graphs of the equation 5x – y = 5 and 3x – y = 3. Determine the coordinates of the vertices of the triangle formed by these lines and the y-axis.

Solution :  The given linear equations are 5x – y = 5           ………(1) 3x – y = 3            ………(2) From equation (1),   y = 5x – 5 When x = 1, y = 5 – 5 = 0 When x = 2, y = 10 – …

In $$\triangle$$ ABC, $$\angle$$C = 3$$\angle$$B = 2($$\angle$$A + $$\angle$$B). Find the three angles of the triangle.

Solution : Given,  $$\angle$$C = 2($$\angle$$A + $$\angle$$B)           …….(1) Adding 2$$\angle$$C on both sides in equation (1), we get $$\angle$$C + 2$$\angle$$C = 2($$\angle$$A + $$\angle$$B)  + 2$$\angle$$C $$\implies$$  3$$\angle$$C = 2($$\angle$$A + $$\angle$$B + $$\angle$$C) Since, $$\angle$$A + $$\angle$$B + $$\angle$$C = 180 degrees $$\implies$$  $$\angle$$C = $$2\over 3$$ $$\times$$ …

The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.

Solution : Let students in each row be x. And Let number of rows be y. then, total number of students = xy Case 1 : When 3 students are extra in a row, then number of rows becomes (y – 1) xy = (x + 3)(y – 1) $$\implies$$  x – 3y + 3 …

A train covered a certain distance at a uniform speed. If the train would have been 10 km/hr faster, it would have taken 2 hours less than the scheduled time. And if the train were slower by 10 km/hr, it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.

Solution : Let x km/hr  be the original speed of the train and y hrs be the time taken by the train to complete the journey. Then, Distance covered = xy km Case 1 : When  speed = (x + 10) km/hr time taken = (y – 2) hr Distance = (x + 10) (y …

One says, “Give me a hundred, friend ! I shall then become twice as rich as you.” The other replies, “If you give me ten, I shall be six times as rich as you.” Tell me what is the amount of their (respectively) capital ?

Solution : Let two friends named as P and Q. Let P has Rs x and Q has Rs y. According to Question, x + 100 = 2(y – 100) $$\implies$$  x – 2y + 300 = 0           ………(1) y + 10 = 6(x – 10) $$\implies$$  6x – y – …

The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The age of Cathy and Dharam are differ by 30 years. Find the ages of Ani and Biju.

Solution : Let x and y be the ages of Ani and Biju respectively. Then, According to Question, x + y = $$\pm 3$$ Dharam’s age = 2x,  and  Cathy’s age = $$y\over 2$$ Clearly, Dharam is older than Cathy. So, 2x – $$y\over 2$$ = 30 $$\implies$$  4x – y = 60 Thus, we …

Formulate the following problem as a pair of equations, and hence find their solutions

Question : Formulate the following problem as a pair of equations, and hence find their solutions (i)  Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours, find her speed of rowing in still water and the speed of the current. (ii)  2 women and 5 men can together …

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

Question : Solve the following pairs of equations by reducing them to a pair of linear of linear equations : (i)  $$1\over 2x$$ + $$1\over 3y$$ = 2     and    $$1\over 3x$$ + $$1\over 2y$$ = 2 (ii)  $$2\over \sqrt{x}$$ + $$3\over \sqrt{y}$$ = 2       and    $$4\over \sqrt{x}$$ – $$9\over \sqrt{y}$$ …