2. On comparing the ratios a1/a2 , b1/b2 , c1/c2 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 (ii) 9x + 3y + 12 = 0 18x + 6y + 24 = 0 (iii) 6x – 3y + 10 = 0 2x – y + 9 = 0
On comparing the ratio, (a1/a2) , (b1/b2) , (c1/c2) 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
Answers
Answer:
q1
(i) Given expressions;
5x−4y+8 = 0
7x+6y−9 = 0
Comparing these equations with a1x+b1y+c1 = 0
And a2x+b2y+c2 = 0
We get,
a1 = 5, b1 = -4, c1 = 8
a2 = 7, b2 = 6, c2 = -9
(a1/a2) = 5/7
(b1/b2) = -4/6 = -2/3
(c1/c2) = 8/-9
Since, (a1/a2) ≠ (b1/b2)
So, the pairs of equations given in the question have a unique solution and the lines cross each other at exactly one point.
(ii) Given expressions;
9x + 3y + 12 = 0
18x + 6y + 24 = 0
Comparing these equations with a1x+b1y+c1 = 0
And a2x+b2y+c2 = 0
We get,
a1 = 9, b1 = 3, c1 = 12
a2 = 18, b2 = 6, c2 = 24
(a1/a2) = 9/18 = 1/2
(b1/b2) = 3/6 = 1/2
(c1/c2) = 12/24 = 1/2
Since (a1/a2) = (b1/b2) = (c1/c2)
So, the pairs of equations given in the question have infinite possible solutions and the lines are coincident.
(iii) Given Expressions;
6x – 3y + 10 = 0
2x – y + 9 = 0
Comparing these equations with a1x+b1y+c1 = 0
And a2x+b2y+c2 = 0
We get,
a1 = 6, b1 = -3, c1 = 10
a2 = 2, b2 = -1, c2 = 9
(a1/a2) = 6/2 = 3/1
(b1/b2) = -3/-1 = 3/1
(c1/c2) = 10/9
Since (a1/a2) = (b1/b2) ≠ (c1/c2)
So, the pairs of equations given in the question are parallel to each other and the lines never intersect each other at any point and there is no possible solution for the given pair of equation
q2
Comparing these equations with a1x+b1y+c1 = 0
And a2x+b2y+c2 = 0
We get,
a1 = 3, b1 = 2, c1 = -5
a2 = 2, b2 = -3, c2 = -7
(a1/a2) = 3/2
(b1/b2) = 2/-3
(c1/c2) = -5/-7 = 5/7
Since, (a1/a2) ≠ (b1/b2)
So, the given equations intersect each other at one point and they have only one possible solution. The equations are consistent.
(ii) Given 2x – 3y = 8 and 4x – 6y = 9
Therefore,
a1 = 2, b1 = -3, c1 = -8
a2 = 4, b2 = -6, c2 = -9
(a1/a2) = 2/4 = 1/2
(b1/b2) = -3/-6 = 1/2
(c1/c2) = -8/-9 = 8/9
Since , (a1/a2) = (b1/b2) ≠ (c1/c2)
So, the equations are parallel to each other and they have no possible solution. Hence, the equations are inconsistent.
(iii)Given (3/2)x + (5/3)y = 7 and 9x – 10y = 14
Therefore,
a1 = 3/2, b1 = 5/3, c1 = -7
a2 = 9, b2 = -10, c2 = -14
(a1/a2) = 3/(2×9) = 1/6
(b1/b2) = 5/(3× -10)= -1/6
(c1/c2) = -7/-14 = 1/2
Since, (a1/a2) ≠ (b1/b2)
So, the equations are intersecting each other at one point and they have only one possible solution. Hence, the equations are consistent.
(iv) Given, 5x – 3y = 11 and – 10x + 6y = –22
Therefore,
a1 = 5, b1 = -3, c1 = -11
a2 = -10, b2 = 6, c2 = 22
(a1/a2) = 5/(-10) = -5/10 = -1/2
(b1/b2) = -3/6 = -1/2
(c1/c2) = -11/22 = -1/2
Since (a1/a2) = (b1/b2) = (c1/c2)
These linear equations are coincident lines and have infinite number of possible solutions. Hence, the equations are consistent.
(v)Given, (4/3)x +2y = 8 and 2x + 3y = 12
a1 = 4/3 , b1= 2 , c1 = -8
a2 = 2, b2 = 3 , c2 = -12
(a1/a2) = 4/(3×2)= 4/6 = 2/3
(b1/b2) = 2/3
(c1/c2) = -8/-12 = 2/3
Since (a1/a2) = (b1/b2) = (c1/c2)
These linear equations are coincident lines and have infinite number of possible solutions. Hence, the equations are consistent.
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