Question

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On comparing the ratios a1/a2, b1/b2, and 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​

Answer 1

Answer:

$\displaystyle\huge\red{\underline{\underline{SoLuTiOn}}}$

(i) Given : 3x + 2y = 5 or 3x + 2y – 5 = 0

and 2x – 3y = 7 or 2x – 3y – 7 = 0

Comparing the above 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 the lines intersect each other at a point and have only one possible solution.

Hence, 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

Therefore, the lines are parallel to each other and they have no possible solution. Hence, the equations are inconsistent.

Answer 2

$\star\:\:\:\bf\large\underline\blue{Question:-}$

• Check whether the following pair of linear equations are consistent or inconsistent.

$\star\:\:\:\bf\large\underline\blue{Solution:-}$

$\sf\large Number\:1:-$

First of all, convert the given equation in the form of

$\sf a_{1}x + b_{1}y + c_{1} = 0 \: ...(i)$

And

$\sf a_{2}x + b_{2}y + c_{2} = 0 \: ...(ii)$

So, we get,

$\sf3x + 2y - 5 = 0$

And

$\sf 2x - 3y - 7 = 0$

Comparing with equations (i) and (ii), we get,

$\sf a_{1} = 3 \: and \: a_{2} = 2$

$\sf b_{1} = 2 \: and \: b_{2} = - 3$

$\sf c_{1} = - 5 \: and \: c_{2} = - 7$

Therefore, we get,

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

Hence, lines are consistent.

$\sf\large Number\:2:-$

Again, convert the equations in standard form.

We get,

$\sf2x - 3y - 8 = 0$

And,

$\sf4x - 6y - 9 = 0$

Now,

$\sf a_{1} = 2 \: and \: a_{2} = 4$

$\sf b_{1} = - 3 \: and \: b_{2} = - 6$

$\sf c_{1} = - 8 \: and \: c_{2} = - 9$

We get,

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

Hence, lines are inconsistent.