Question

Represent the following pair of linear equations graphically and hence comment on the condition of consistency of this pair:

x – 5y = 6; 2x – 10y = 12

Answer 1

1. Identify points for the first equation

For the equation x – 5y = 6

If y = 0, then x – 5(0) = 6

⇒ x = 6

Point: (6, 0)

If y = –2, then x – 5(–2) = 6

⇒ x + 10 = 6

⇒ x = –4

Point: (–4, –2)

If x = 11, then 11 – 5y = 6

⇒ –5y = –5

⇒ y = 1

Point: (11, 1)

2. Identify points for the second equation

For the equation 2x – 10y = 12:

If y = 0, then 2x – 10(0) = 12

⇒ 2x = 12

⇒ x = 6

Point: (6, 0)

If x = 1, then 2(1) – 10y = 12

⇒ 2 – 10y = 12

⇒ –10y = 10

⇒ y = –1

Notice that dividing the entire second equation by 2 gives x – 5y = 6, which is identical to the first equation.

3. Represent the equations graphically

Since both equations simplify to the same linear relationship, plotting them results in a single line on the coordinate plane.

4. Comment on the condition of consistency

To determine consistency algebraically, compare the ratios of the coefficients `(a_1)/(a_2), (b_1)/(b_2)` and `(c_1)/(c_2)` for equations in the form a₁x + b₁y = c₁ and a₂x + b₂y = c₂:

`(a_1)/(a_2) = 1/2`

`(b_1)/(b_2) = (-5)/(-10) = 1/2`

`(c_1)/(c_2) = 6/12 = 1/2`

Since `(a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2)`, the lines are coincident. This means every point on the line is a solution, resulting in infinitely many solutions. Therefore, the system is consistent and dependent.