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Determine by drawing the graph, whether the following system of linear equations has a unique solution or not $3 x − 2 y = 12; x − 23 y − 4 = 0$

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unique solution

no common solution

infinitely many solutions

none of the above

answer is A.

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Detailed Solution

The given pair of linear equations is

3 x + 2 y = 12

and

5 x − 2 y = 4

.
To solve the equations graphically we have to get the coordinated for each equation.
For the linear equation

3 x + 2 y = 12

,
Putting

x = 0

in the equation

3 x + 2 y = 12

to obtain the value of

y

⇒ 3 × 0 + 2 y = 12 ⇒ 2 y = 12 ⇒ y = 122 ⇒ y = 6

Putting x=2 in the equation

3 x + 2 y = 12

to obtain the value of

y

⇒ 3 × 2 + 2 y = 12 ⇒ 2 y = 12 − 6 ⇒ y = 62 ⇒ y = 3

Putting

x = 4

in the equation

3 x + 2 y = 12

to obtain the value of

y

⇒ 3 × 4 + 2 y = 12 ⇒ 2 y = 12 − 12 ⇒ y = 0

So, the coordinates of the line equation

3 x + 2 y = 12

are

0, 6, 2, 3

and

4, 0

.
For the second line equation

5 x − 2 y = 4

,
Putting

x = 0

in the equation

5 x − 2 y = 4

to obtain the value of

y

⇒ 50 − 2 y = 4 ⇒ − 2 y = 4 ⇒ y = − 2

Putting x=2 in the equation

5 x − 2 y = 4

to obtain the value of

y

\Rightarrow 5 (2) - 2 y = 4

\Rightarrow 10 - 2 y = 4

\Rightarrow 6 = 2 y

\Rightarrow y = 3

Putting

x = 4

in the equation

5 x − 2 y = 4

to obtain the value of

y

⇒ 54 − 2 y = 4 ⇒ 2 y = 20 − 4 ⇒ y = 8

So, the coordinate of the line equation

5 x − 2 y = 4

are

0, − 2, 0.8, 0

and

4, 8

.
Plot the graph of the line

3 x + 2 y = 12

using the table below:

X	0	2	4
Y	6	3	0

And, plot the graph of line

5 x − 2 y = 4

using the table below:

X	0	2	4
Y	-2	3	8

The graph of the lines is represented as,
Question Image

From the graph we can see that the lines

3 x + 2 y = 12

and

5 x − 2 y = 4

are intersecting at the point

2, 3

hence, the system of equations have a unique solution.
Hence, the correct option is 1.

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Determine by drawing the graph, whether the following system of linear equations has a unique solution or not 3x−2y=12;x− 2 3 y−4=0

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Determine by drawing the graph, whether the following system of linear equations has a unique solution or not $3 x − 2 y = 12; x − 23 y − 4 = 0$