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# By the graphical method, find whether the following pair of equations are consistent or not. If consistent, solve them.

$ x+y=3 $

$ 3 x+3 y=9 $

Given:

Pair of linear equations:

\( x+y=3 \)

\( 3 x+3 y=9 \)

To do:

We have to find whether the given pair of linear equations is consistent /inconsistent. If consistent, then obtain the solution graphically.

Solution:

$x+y-3=0$.........(i)

$3x+3y-9=0$.........(ii)

Here,

$a_{1}=1, b_{1}=1, c_{1}=-3$

$a_{2}=3, b_{2}=3, c_{2}=-9$

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

$\frac{b_{1}}{b_{2}}=\frac{1}{3}$

$\frac{c_{1}}{c_{2}}=\frac{-3}{-9}$

$=\frac{1}{3}$

$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}=\frac{c_1}{c_2}$

This implies,

The given pair of lines is coincident and therefore consistent.

For equation (i),

$y=3-x$

$x$ | 0 | $3$ |

$y$ | $3$ | $0$ |

Plot point $( 0,\ 3)$ and $(3,\ 0)$ on a graph and join then to get equation $x+y=3$

For equation (ii),

$3y=9-3x$

$y=3-x$

$x$ | $0$ | 3 |

$y$ | $3$ | 0 |

Plot points $(0,\ 3)$ and $(3,\ 0)$ on a graph and join them to get equation $3x+3y=9$

There are infinite solutions as the lines are coincident.

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