Question

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

x + y = 3, 3x + 3y = 9

Answer 1

Given pair of equation are

x + y = 3

⇒ x + y – 3 = 0   ......(i)

And 3x + 3y = 9

⇒ 3x + 3y – 9 = 0   .......(ii)

On comparing with ax + by + c = 0, we get

a₁ = 1, b₁ = 1 and c₁ = –3  ......[From (i)]

a₂ = 3, b₂ = 3 and c₂ = –9  ......[From (ii)]

Here, `a_1/a_2 = 1/3`,

`b_1/b_2 = 1/3`

And `c_1/c_2 = (-3)/(-9) = 1/3`

⇒ `a_1/a_2 = b_1/b_2 = c_1/c_2`

So, the given pair of lines is coincident.

Therefore, these lines have infinitely many solutions.

Hence, the given pair of linear equations are consistent.

Now, x + y = 3

⇒ y = 3 – x

x	0	3	2
y	3	0	1

And 3x + 3y = 9

⇒ 3y = 9 – 3x

⇒ y = `(9 - 3x)/3`

x	0	1	3
y	3	2	0

Plotting the points we get the graph of lines.

We observe that the lines represented by (i) and (ii) are coincident.