3. Pair of Linear Equations In Two Variables

3.1 Introduction

3.2 Graphical Method of Solution of a pair of Liniar Eqations

Example 1:

Check graphically wheather the pair of equations

x + 3y = 6 and 2x -3y = 12

os consitent. If so solve them graphically.

Example 2:

Graphically, find wheather the following pair of equations has no solution, unique solution of inifinitely many solutions:

(1) 5x-8y+1=0

(2) 3x - 2/5y + 3/5 = 0

Example 3:

Champa went to a 'Sale' to purchase some pants and skirts. When her friends aked 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 numner of skirts is four less than four times the number of pants purchased". Help her friends to find how many panrs and skirts Champa bought.

EXERCISE 3.1

1. Form the pair of liniear 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 who took part in the quiz.

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

2. On comparing the ratios a1/a2,b1/b2 and c1/c2, find out whether the lines representing the following pairs of liniar eqations intersect at a point, are parallel or coincident:

(i) 5x - 4y + 8 = 0, 7x + 6y - 9 = 0 (ii) 9x + 3y + 12 = 0 (iii) 6x - 3y + 10 = 0, 2x - y + 9 = 0

3. On comparing the ratios a1/a2,b1/b2 and c1/c2, find out whether the following pair of liniar equations are consistent, or inconsitent.

(i) 3x + 2y = 5; 2x -3y = 7 (ii) 2x - 3y = 8; 4x - 6y = 9 (iii) 3/2x + 5/3y = 7; 9x - 10y = 14 (iv) 5x - 3y = 11; 10x + 6y = -22 (v) 4/3x + 2y = 8; 2x + 3y = 12

4. Which of the following pairs of liniar equations are consistent/inconsistent? If consistent, obtain the solution graphically:

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

(ii) x - y = 8, 3x - 3y = 16

(iii) 2x + y - 6 = 0, x -2y - 4 = 0

(iv) 2x - 2y -2 = 0, 4x - 4y - 5 = 0

5. Half the perimeter of a rectangular garden, whose length is m amore than its width, is 36m. Find the dimensions of the garden.

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

(i) intersecting lines (ii) parallel lines (iii) coincident lines

7. 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.

3.3 Algebric Method of Solving of Linear Equations

3.3.1 Substitution Method

We shal expalin the method of substitution by taking some examples.

Example 4

Solve the following pair of equations by substitution method:

7x -15y = 2

x = 2y = 3

Example 5:

Solve the following question - aftab tells his doughter, "Seven years ago, I was seven times as old as you will be." (Isn't this interesting?) Represent this situation algebrically and graphically by the method by the method of substitution.

Example 6:

In a shop the cost of 2 pencils and 3 erasers is ₹ 9 and the cost of pencil and 6 erasers is ₹ 18. Find the cost of each pencil and each eraser.

Example 7:

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

EXERCISE 3.2

1. Solve the following pair of liniar equations by the substiturion method.

(i) x + y = 1, x - y = 4 (ii) s - t = 3, s/3 + t/2 = 6 (iii) 3x - y = 3, 9x - 3y = 9 (iv) 3x/2 - 5y/3 = -2 (v) √2x + √3y = 0, √3x - √8y = 0 (vi) 3x/2 - 5y/3 = -2, x/3 + y/2 = 13/6

2. Solve 2x + 3y = 11 and 2x - y = - 24 and hence find the value of 'm' for which y = mx + 3.

3. Form the pair fo linear equations for the following problems and find their solution by substitution method.

(i) The difference between two numbers is 26 and one number is three times the other. Find them.

(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

(iii) The coach of a cricket team buys 7 bats and 6 balls for ₹ 3800. Later, she buys 3 bats and 5 balls for ₹ 1750. Find the cost of each bat and each ball.

(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is ` 105 and for a journey of 15 km, the charge paid is ` 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?

A fraction becomes 9/11, if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction.

3.3.2 Elimination Method

Example 8:

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 &8377; 2000 per month, find their monthly incomes.

Example 9:

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

2x + 3y = 8 (1)

4x + 6y = 7 (2)

Example 10:

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?

EXERCISE 3.3

1. Solve the following pair of linear equations by the elimination method and the substitution method :

(i) x + y = 5 and 2x – 3y = 4

(ii) 3x + 4y = 10 and 2x – 2y = 2

(iii) 3x – 5y – 4 = 0 and 9x = 2y + 7

(iv) x/2 + 2y/3 = -1 and x - y/3 = 3

Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method :

(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator. What is the fraction?

(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

(iv) Meena went to a bank to withdraw &8377; 2000. She asked the cashier to give her &8377; 50 and &8377; 100 notes only. Meena got 25 notes in all. Find how many notes of &8377; 50 and &8377; 100 she received.

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹ 27 for a book kept for seven days, while Susy paid ₹ 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.