Which of the following condition is true if the system of equations below is shown to be consistent and dependent?
$a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$

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

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

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

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

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Solution

The correct option is B $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
If the system of equations are consistent, then there can be exactly one solution or infinite solutions.
The solutions are infinite when they are dependent.

We know that, for two linear equations
$a_{1} x + b_{1} = c_{1}$ and $a_{2} x + b_{2} y = c_{2}$
If $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ , the system is consistent and has infinitely many solutions.

Hence, a system of equations will be consistent and dependent if $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ .

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Which of the following condition is true if the system of equations below is shown to be consistent and dependent?a1x+b1y+c1=0,a2x+b2y+c2=0

Which of the following condition is true if the system of equations below is shown to be consistent and dependent?
$a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$