Question

a pair of linear equation a1x+b1y+c1=0; and a2x+b2y+c2=0 is said to be inconsistent,if (a) a1/a2not equal to b1/b2. (b) a1/a2=b1/b2not equal to c1/c2. (c) a1/a2=b1/b2=c1/c2. (d) a1/a2 not equal to c1/c2. please give answer with solution and exact option​

Answer 1

Given a pair of linear equations:

$a_1x+b_1y+c_1=0\\a_2x+b_2y+c_2=0$

The two lines have a solution when.

$1.\ \dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}$

For example, the following pair will have exactly one solution:

$x+2y+3=0\\2x+y+9=0$

Let us calculate the ratio of $a_1, a_2\ and\ b_1, b_2$:

$\dfrac{a_1}{a_2} = \dfrac{1}{2}$

$\dfrac{b_1}{b_2} = \dfrac{2}{1}$

Here, the ratio is not equal:

$\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}$

Hence, the pair of linear equation is consistent with exactly one solution.

$2.\ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$

If the ratio of $a_1: a_2 = b_1: b_2 = c_1:c_2$ (equal), it will have consistent solution with infinite solutions.

Actually, the lines are identical.

For example:

$x+2y+3=0\\2x+4y+6=0$

Here,

$\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2} = \dfrac{1}{2}$

So, the pair of linear equations is consistent with infinite solutions.

$3.\ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq \dfrac{c_1}{c_2}$

That means, the ratio of $a_1, a_2\ and\ b_1, b_2$ are same but not equal to ratio $c_1:c_2$. In this case, the lines will be parallel to each other and there will no or the pair of linear equation is inconsistent.

For example:

$x+2y+9=0\\2x+4y+6=0$

Here, the ratio is:

$\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{1}{2}\\\dfrac{c_1}{c_2} = \dfrac{9}{6} = \dfrac{3}{2}\\\therefore \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}$

Hence, correct answer is option

$(B)\ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq \dfrac{c_1}{c_2}$

Answer 2

Answer:

the lines represented by the equation a1 X + b1 y+ c1 = 0 and a2x + b2 Y + c2 = 0 and coincident if: a) a1/a2≠b1/b2.b)a1/a2=b1/b2=c1/2. c) a1/a2=b2/b2≠c1/c2. d) a1/a2≠b1/b2≠c1/c2​

Step-by-step explanation: