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    Exercise 10.3.1

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6th class > > Exercise 10.3.1

Exercise 10.3.1

1. 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.

The total number of students is 10 : x+y=10.
The number of girls is 4 more than the number of boys : y=x+4.
Equation 1.x+y=10 x=0,y=10.point(0,10),x=10,y=0.point(,).
Equation 2.y=x+4 x=0,y=4.point(0,4),x=6,y=10.point(,)

(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

5 pencils and 7 pens cost \Rs50 : 5x+7y=50, (Equation 1)
7 pencils and 5 pens cost \Rs46 : 7x+5y=46, (Equation 2)
Multiply Equation 1 by 5 : 5(5x+7y)=5(50) : 25x+35y=, (Equation 3).
Multiply Equation 2 by 7 : 7(7x+5y)=7(46) : 49x+35y=, (Equation 3).
Subtract Equation 3 from Equation 4 : (49x + 35y)-(25x + 35y) = 322 - 250.
: 24x=72 : x=.
Substitute x into Equation 1 : 5(3)+7y=50 : +7y=50 : y = .

2. On comparing the ratios a1a2, b1b2and c1c2, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.

(i) 5x – 4y + 8 = 0,7x + 6y – 9 = 0

For the first pair of equations: 5x - 4y + 8 = 0 : a1 = ,b1 = ,c1 = .
7x + 6y – 9 = 0 : a2 = , b2 = ,c2 = .
Compare the rations:a1a2 = 57, b1b2 = −23,c1c2 = −89.
The lines at a point.

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

For the first pair of equations: 9x + 3y + 12 = 0 : a1 = ,b1 = ,c1 = .
18x + 6y + 24 = 0 : a2 = , b2 = ,c2 = .
Compare the rations:a1a2 = 12, b1b2 = 12,c1c2 = 12.
The lines are .

(iii) 6x – 3y + 10 = 0,2x – y + 9 = 0

For the first pair of equations: 6x – 3y + 10 = 0 : a1 = ,b1 = ,c1 = .
2x – y + 9 = 0 : a2 = , b2 = ,c2 = .
Compare the rations:a1a2 = 3, b1b2 = 3,c1c2 = 109.
The lines are .

3. On comparing the ratios a1a2, b1b2and c1c2, find out whether the following pair of linear equations are consistent, or inconsistent.

Hint: 1.The system of equations is consistent and independent (intersecting lines).

2.The system of equations is consistent and dependent (coincident lines).

3.The system of equations is inconsistent (parallel lines).

(i) 3x + 2y = 5 ; 2x – 3y = 7

Lets analyze each pair of equations: 3x + 2y = 5 : a1 = ,b1 = ,c1 = .
2x – 3y = 7 : a2 = , b2 = ,c2 = .
Compare the rations:a1a2 = 32, b1b2 = 2−3,c1c2 = 57.
The system is and independent.

(ii) 2x – 3y = 8 ; 4x – 6y = 9

For the first pair of equations: 2x – 3y = 8 : a1 = ,b1 = ,c1 = .
4x – 6y = 9 : a2 = , b2 = ,c2 = .
Compare the rations:a1a2 = 12, b1b2 = 12,c1c2 = 89.
The system is .

(iii) 32x+53y=7 ; 9x – 10y = 14

For the first pair of equations: 32x+53y=7 : a1 = /,b1 = /,c1 = .
9x – 10y = 14 : a2 = , b2 = ,c2 = .
Compare the rations:a1a2 = 16, b1b2 = −16.
The system is and independent.

(iv) 5x – 3y = 11 ; – 10x + 6y = –22

For the first pair of equations: 5x – 3y = 11 : a1 = ,b1 = ,c1 = .
10x + 6y = –22 : a2 = , b2 = ,c2 = .
Compare the rations:a1a2 = −12, b1b2 = −12,c1c2 = −12.
The system is and dependent.

(v)43x+2y=8 ; 2x + 3y = 12

For the first pair of equations: 43x+2y=8 : a1 = /,b1 = ,c1 = .
2x + 3y = 12 : a2 = , b2 = ,c2 = .
Compare the rations:a1a2 = 23, b1b2 = 23,c1c2 = 23.
The system is .

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

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

The equations: x + y = 5, 2x + 2y = 10
Compare the rations:a1a2 = /2, b1b2 = 1/,c1c2 = /.
The pair of linear equations is .

(ii) x – y = 8, 3x – 3y = 16

The equations: x – y = 8, 3x – 3y = 16
Compare the rations:a1a2 = 1/, b1b2 = /,c1c2 = 1/.
The pair of linear equations is .

(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0

The equations: 2x + y – 6 = 0, 4x – 2y – 4 = 0
Compare the rations:a1a2 = /, b1b2 = /2,c1c2 = /s.
The pair of linear equations is .

(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0

The equations: 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
Compare the rations:a1a2 =/, b1b2 = /,c1c2 = 2/.
The pair of linear equations is .

5. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

Let the width of the garden be x and length be y.

According to the question,

1.y - x = 4 (1) : y - x = 4 :y = x + 4

x0812
y41216

2.y + x = 36 (2) : y + x = 36

x03616
y36020

6. 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.

Given Equation:2x+3y-8=0

(i) intersecting lines

a1 = , b = , c1 = -8; a2, b2, c2 = 1.

Thus,It is x + y + 1 = 0.

(ii) parallel lines

a1 = 2, b = , c1 = ; a2 = 4, b2 = 6, c2 = 1.

Thus,It is 4x + 6y + 1 = 0.

(iii) coincident lines

a1 = 2, b = , c1 = ; a2 = 4, b2 = 6, c2 = -16.

Thus,It is 4x + 6y + 1 = 0.

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.

x - y + 1 = 0 : x = y - 1

x012
y123

3x + 2y - 12 = 0 : x=12−2y3

x420
y036