Become a master of Pair of Linear Equations in Two Variables

Method to Write a Given Statement as a Pair of Linear Equations in Two Variables (Algebraic representation)

Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheel and play Hoopla (a game in which you throw a ring on the items kept in a stall, and if the ring covers any object completely, you get it). The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. If each ride costs ₹3, and a game of Hoopla costs ₹4, how would you find out the number of rides she had and how many times she played Hoopla, provided she spent ₹20. Represent this situation algebraically .

Seven times a two digit number is equal to four times the number obtained by reversing the order of its digits. If the difference between the digits is 3.Represent this situation algebraically .

The ratio of incomes of two persons is 11 : 7 and the ratio of their expenditures is 9 : 5. If each of them manages to save ₹400 per month. Represent this situation algebraically.

Five years ago, sagar was twice as old as Tiru. After 10 years Sagar's age will be ten years more than Tiru's age. To find their present ages form system of Linear Equations.

Method to write a given statement as a pair of L.E.s in 2 variables(Algebraically) and representing graphically (Geometrically).

Graphical Method of Solution of a Pair of Linear Equations

Romila went to a stationery shop and purchased 2 pencils and3 erasers for ₹9. Her friend Sonali saw the new variety of pencils and erasers with Romila, and she also bought 4 pencils and 6 erasers of the same kind for₹18. Represent this situation algebraically and graphically.

Two rails are represented by the equations x + 2y – 4 = 0 and 2x+ 4y – 12 = 0. Represent this situation geometrically.

Champa went to a ‘Sale’ to purchase some pants and skirts. When her friends asked her how many of each she had bought, she answered, “The number of skirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased”. Help her friends to find how many pants and skirts Champa bought.

Behaviour of lines representing a pair of linear equations (Checking consistency).

On comparing the ratios a_1/a_2 ,b_1/b_2  and c_1/c_2 find out whether the lines representing the following pairs of linear equations intersect at a point or parallel or coincide:

3x - y = 7

2x + 5y + 1 = 0

Check graphically whether the pair of equations x + 3y = 6 and 2x – 3y = 12 is consistent. If so, solve them graphically.

Graphically, find whether the following pair of equations has no solution, unique solution or infinitely many solutions:

5x – 8y + 1 = 0

3x - (24/5)y + 3/5= 0

For which values of p does the pair of equations

given below has unique solution?

4x + py + 8 = 0

2x + 2y + 2 = 0

For what values of k will the following pair of

linear equations have infinitely many solutions?

kx + 3y – (k – 3) = 0

12x + ky – k = 0

Algebraic Methods of Solving a Pair of Linear Equations: Substitution Method

Solve the following pair of equations by substitution method:

7x – 15y = 2

x + 2y = 3

Two rails are represented by the equations x + 2y – 4 = 0 and 2x + 4y – 12 = 0. Will the rails cross each other?

Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be." then find their present ages?

Romila went to a stationery shop and purchased 2 pencils and 3 erasers for ₹9. Her friend Sonali saw the new variety of pencils and erasers with Romila, and she also bought 4 pencils and 6 erasers of the same kind for ₹18. Find the cost of each pencil and each eraser.

Algebraic Methods of Solving a Pair of Linear Equations:

Elimination Method

Solve the following pair of linear equations by the elimination method: 3x + 4y = 10 and 2x – 2y = 2.

Use elimination method to find all possible solutions of the following pair of linear equations:

2x + 3y = 8

4x + 6y = 7

The ratio of incomes of two persons is 9: 7 and the ratio of their expenditures is 4 : 3. If each of them manages to save₹2000 per month, find their monthly incomes.

The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2,find the number. How many such numbers are there?

Algebraic Methods of Solving a Pair of Linear Equations:

Cross-Multiplication Method

Solve using cross multiplication method:
5x + 4y - 4 = 0
x - 12y - 20 = 0

Solve the following pair of linear equations by cross multiplication method:

x + 2y = 2
x - 3 y = 7

From a bus stand in Bangalore, if we buy 2 tickets to Malleswaram and 3 tickets to Yeshwanthpur, the total cost is ₹46; but if we buy 3 tickets to Malleswaram and 5 tickets to Yeshwanthpur the total cost is ₹74. Find the fares from the bus stand to Malleswaram and to Yeshwanthpur.

Equations Reducible to a Pair of Linear Equations in Two Variables

Solve the pair of equations:

2/x+ 3/y= 13;

5/x- 4/y= - 2

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

5/(x - 1) + 1/(y - 2) = 2;

6/(x - 1) - 3/(y - 2) = 1

A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km down-stream. Determine the speed of the stream and that of the boat in still water.

Aftab tells his daughter, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be." (Isn't this interesting?) Represent this situation algebraically and graphically.

The coach of a cricket team buys 3 bats and 6 balls for Rs 3900. Later, she buys another bat and 3 more balls of the same kind for Rs 1300. Represent this situation algebraically and graphically.

The cost of 2 kg of apples and 1 kg of grapes on a day was found to be Rs 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is Rs 300. Represent the situation algebraically and geometrically.

Form the pair of linear equations in the following problems, and find their solutions graphically.

(i) 10 students of class X took part in a mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

Form the pair of linear equations in the following problems, and find their solutions graphically

(ii) 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs. 46. Find the cost of one pencil and that of one pen.

On comparing the ratios a_1/a_2  = b_1/b_2  and c_1/c_2  find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide:

(i) 5x − 4y + 8 = 0;  7x + 6y - 9 = 0

(ii)9x + 3y + 12 = 0;  18x + 6y + 24 = 0

(iii) 6x − 3y + 10 = 0;  2x – y + 9 = 0

On comparing the ratios a_1/a_2  = b_1/b_2  and c_1/c_2 , find out whether the following pair of linear equations are consistent, or inconsistent.

(i) 3x + 2y = 5, 2x − 3y = 7

(ii) 2x − 3y = 8, 4x − 6y = 9

(iii) (3/2)x + (5/3)y = 7, 9x − 10y = 14

(iv) 5x − 3y = 11, −10x + 6y = −22

(v) (4/3)x + 2y = 8, 2x + 3y = 12

Which of the following pairs of linear equations are consistent/ inconsistent? If consistent, obtain the solution graphically

(i) x + y = 5, 2x + 2y = 10

Which of the following pairs of linear equations are consistent/ inconsistent? If consistent, obtain the solution graphically

(ii) x – y = 8, 3x − 3y = 16

Which of the following pairs of linear equations are consistent/ inconsistent? If consistent, obtain the solution graphically

(iii) 2x + y - 6 = 0, 4x − 2y - 4 = 0

Which of the following pairs of linear equations are consistent/ inconsistent? If consistent, obtain the solution graphically

(iv) 2x − 2y – 2 = 0, 4x − 4y – 5 = 0

Half the perimeter of a rectangle garden, whose length is 4 m more than its width, is 36 m. find the dimensions of the garden.

Given the linear equation (2x + 3y – 8 = 0), write another linear equation in two variables such that the geometrical representation of the pair so formed is:

(i). Intersecting lines

(ii). Parallel lines

(iii). Coincident lines

Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.