Question

is the pair of equations X+2y-3=0; 6y+3x-9=0 consistent? Justify your answer.​

Answer 1

Answer:

x+2y-3=0; 6y+3x-9=0

compare first equation with a1x+b1y+c=0

compare second equation with a2x+b2y+c=0

1/6≠2/3

a1/a2≠b1/b2

it is intersecting lines, it has unique (one) solution

so it is consistent term

Answer 2

Answer:-

Yes the given pair of equations are consistent i.e. they have one or more common solutuons for x and y.

Explanation:-

Given Equations:-

• $\tt\implies x+2y-3=0 -- (1).$

• $\tt\implies 3x+6y-9=0 --(2).$

To Find:-

• If the pair of equations are consistent or not.

Now,

Let us first understand the meaning of consistent pair of linear equations.

$\bf\small\bigstar{\underline{\underline{\blue{\textbf{ \:Consistent Pair Of Linear Equations}:-}}}}$

• The pair of equations having one or more than one common solution (having common values of unknown variables) are called consistent pair of linear equations.

Things To Know:-

Let us suppose that we have two pair of linear equations:-

$\\ \tt\mapsto a_1x + b_1y + c_1 = 0 - - (1). \\ \\ \tt\mapsto a_2x + b_2y + c_2 = 0 - - (2).$

Then we have,

$\tiny{\bf\bullet\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}\longrightarrow \textbf{Unique Solution, i.e. only 1 common value of x and y.}}$

$\tiny{\bf\bullet\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}\longrightarrow \textbf{Infinite Solutions, i.e. infinite common value of x and y.}}$

$\tiny{\bf\bullet\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}\longrightarrow \textbf{No common Solution, i.e. no common value of x and y is possible.}}$

Solution:-

So In This Case:-

$\\ \tt\mapsto \dfrac{a_1}{a_2} = \frac{1}{ 3 } .$

$\\ \tt\mapsto \dfrac{b_1}{b_2} = \cancel\frac{2}{6} = \frac{1}{3} .$

$\\ \tt\mapsto \dfrac{b_1}{b_2} = \cancel\frac{ - 3}{ - 9} = \frac{1}{3} .$

Therefore here, $\bf\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}.$

So The Given Pair Of Equations Have Infinite Solutions.

And Hence The Given Pair Of Linear Equations Are Consistent.