Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

(i) x + y = 5, 2x + 2y = 10
(ii) x – y = 8, 3x – 3y = 16
(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0

Solution:

(i) Given, x + y = 5 and 2x + 2y = 10

Comparing these equations with a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 we get,

(a1/a2) = 1/2

(b1/b2) = 1/2

(c1/c2) = 1/2

Here, (a1/a2) = (b1/b2) = (c1/c2)

∴The equations are coincident and they have an infinite number of possible solutions.

So, the equations are consistent.

For, x + y = 5 or y = 5 – x

x 0 3 5
y 5 2 0

For 2x + 2y = 10 or y = (10 – 2x)/2

x 0 3 5
y 5 2 0

So, the equations are represented in graphs as follows:

NCERT Solutions Class 10 Maths Ch 3 Ex3.1 Q4 i

From the figure, we can see, that the lines are overlapping each other.

Therefore, the equations have infinite possible solutions.

(ii) Given, x – y = 8 and 3x – 3y = 16

Comparing these equations with a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 we get,

(a1/a2) = 1/3

(b1/b2) = -1/-3 = 1/3

(c1/c2) = 8/16 = 1/2

Here, (a1/a2) = (b1/b2) ≠ (c1/c2)

The equations are parallel to each other and have no solutions. Hence, the pair of linear equations is inconsistent.

(iii) Given, 2x + y – 6 = 0 and 4x – 2y – 4 = 0

Comparing these equations with a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 we get,

(a1/a2) = 2/4 = ½

(b1/b2) = 1/-2

(c1/c2) = -6/-4 = 3/2

Here, (a1/a2) ≠ (b1/b2)

The given linear equations are intersecting each other at one point and have only one solution. Hence, the pair of linear equations is consistent.

Now, for 2x + y – 6 = 0 or y = 6 – 2x

x 0 1 2
y 6 4 2

And for 4x – 2y – 4 = 0 or y = (4x – 4)/2

x 1 2 3
y 0 2 4

So, the equations are represented in graphs as follows:

NCERT Solutions Class 10 Maths Ch3 Ex3.1 4 iii

From the graph, it can be seen that these lines are intersecting each other at only one point,(2,2).

(iv) Given, 2x – 2y – 2 = 0 and 4x – 4y – 5 = 0

Comparing these equations with a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 we get,

(a1/a2) = 2/4 = ½

(b1/b2) = -2/-4 = 1/2

(c1/c2) = 2/5

Here, (a1/a2) = (b1/b2) ≠ (c1/c2)

Thus, these linear equations have parallel and have no possible solutions. Hence, the pair of linear equations are inconsistent.