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Check whether the pair of equations $x + 3 y = 6$ and $2 x - 3 y = 12$ is consistent. If so, solve graphically,

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An Intiative by Sri Chaitanya

x = 6, y = 0

x = 3, y = 2

x = 3, y = 3

x = 1, y = 1

answer is A.

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Detailed Solution

The given system of equation is,

x + 3 y = 6

….(1)

2 x - 3 y = 12

….(2)
The given pair can be written as,

x + 3 y - 6 = 0

and

2 x - 3 y - 12 = 0

Now the system of linear equations

a_{1} x + b_{1} y + c_{1} = 0

…..(1)

a_{2} x {+ b}_{2} y + c_{2} = 0

……(2)
is consistent if

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

.
Here

a_{1} = 1, b_{1} = 3, c_{1} = - 6

….from (1)

and a_{2} = 2, b_{2} = - 3, c_{2} = - 12

…..from (2)
Now ,

\frac{a_{1}}{a_{2}} = \frac{1}{2},

\frac{b_{1}}{b_{2}} = \frac{3}{- 6} = \frac{1}{- 2}

.
Hence for being consistent system of equations the given system must satisfy,

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

.
Now

\frac{1}{2} \neq - \frac{1}{2}

.
Thus, the given pair of equations is consistent.
To solve them graphically we have plotted the equations in the graph and got the figure below.
Question Image

Consider

x + 3 y - 6 = 0

and

x = 0

( i.e.Y axis)
Then putting the value of x = 0 in the equation (1) we gwt,

0 + 3 y - 6 = 0

3 y = 6

⟹ y = 2

Now putting

x = 3

in the equation (1) we get,

3 + 3 y - 6 = 0

3 y = 3

⟹ y = 1

Join the points by drawing a line.
Consider

2 x - 3 y - 12 = 0

x = 0

(i.e. Y axis)

2 (0) - 3 y - 12 = 0

3 y = 12

⟹ y = - 4

Now putting

x = 3

in equation (2) we get,

2 (3) - 3 y - 12 = 0

- 3 y = 6

y = - 2

.
Join the two points by drawing a line.
The given pair of linear equation has unique solution

x = 6

and

y = 0

.
So the correct option is 1.

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