Question

On comparing the ratios a1/a2, b1/b2, and c1/c2 and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point or are parallel or coincide.
(i) 3x - 5y + 8 = 0, 7x+6y - 9 = 0

(ii) 4x + 3y - 7 = 0, 12x +9y = 21

(iii) x - 2y + 5 = 0, 8y - 4x + 20 = 0​

Answer 1

Solution:

(i) The given pair of linear equations is

3x - 5y + 8 = 0,

7x+6y - 9 = 0

On comparing the given equations with standard form of pair of linear equations i.e. $\pink{a_{1}x + b_{1}y + c_{1} = 0}$ and $\red{a_{2}x + b_{2}y + c_{2} = 0}$

, we get

$\green{a_{1} = 3, \: b_{1} = - 5 , \: c_{1} = 8}$ and, $\purple{a_{2} = 7, \: b_{2} = 6 , \: c_{2} = - 9}$

Here,

$\frac{ a_{1} }{a_{2}} = \frac{3}{7}$

$\frac{b_{1}}{b_{2}} = \frac{ - 5}{6}$

$\therefore \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$

The lines representing the given pair of linear equations will intersect at a point.

.

(ii) The given pair of linear equations is

4x + 3y - 7 = 0,

12x +9y - 21 = 0

On comparing the given equations with standard form of pair of linear equations we get,

$\orange{a_{1} = 4, \: b_{1} = 3 , \: c_{1} = - 7}$ and $\blue{a_{2} = 12, \: b_{2} = 9, \: c_{2} = - 21}$

Now,

$\frac{a_{1} }{a_{2}} = \frac{4}{12} =\frac{1}{3},\frac{b_{1} }{b_{2}} = \frac{3}{9} =\frac{1}{3}$ and,

$\frac{c_{1} }{c_{2} }= \frac{ - 7}{ - 12} = \frac{1}{3}$

$\therefore \: \frac{a_{1} }{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{1}}$

Hence,

The lines representing the given pair of linear equations will coincide.

.

(iii) The given pair of linear equations is

x - 2y + 5 = 0,

- 4x + 8y + 20 = 0

On comparing the given equations with standard form of pair of linear equations we get,

$\green{a_{1} = 1, \: b_{1} = - 2, \: c_{1} = 5}$ and $\purple{a_{2} = - 4, \: b_{2} = 8 , \: c_{2} = 20}$

Now,

$\frac{a_{1} }{a_{2}} = \frac{ - 1}{4} , \: \frac{b_{1} }{b_{2}} = \frac{ - 2}{8} , \: \frac{c_{1}}{c_{2}} = \frac{5}{20} = \frac{1}{4}$

So,

$\frac{a_{1}}{a_{2} }= \frac{b_{1}}{b_{2} } \neq \frac{c_{1}}{c_{2} }$

Therefore,

The lines representing the given pair of linear equations are parallel.

Answer 2

(i)3x - 5y + 8 = 0, 7x+6y - 9 = 0

Checking ratios

a1/a2 = 3/7 and b1/b2 = (-5)/7

So,

a1/a2 ≠ b1/b2

The given pair of linear equation will intersect. (one unique solution)

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(ii) 4x + 3y - 7 = 0, 12x +9y - 21 = 0

a1/a2 = b1/a2 = c1/c2 [each 1/3]

So, the lines will coincide. (many solutions)

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(iii) x - 2y + 5 = 0, - 4x + 8y + 20 = 0​

a1/a2 = b1/b2 ≠ c1/c2

The lines will be parallel. (no solution)