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-2x - 3y = 1; 6y + 4x = -2, do these equations represent a pair of coincident lines

Solution:

Given, the pair of equations is

-2x - 3y = 1

6y + 4x = -2

We have to determine if the equations represent a pair of coincident lines.

The equation 6y + 4x = -2 can be written as 4x + 6y = -2.

Here, a₁ = -2, b₁ = -3, c₁ = 1

a₂ = 4, b₂ = 6, c₂ = -2

So, a₁/a₂ = -2/4 = -1/2

b₁/b₂ = -3/6 = -1/2

c₁/c₂ = 1/-2 = -1/2

-1/2 = -1/2 = -1/2

\(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

We know that,

For a pair of linear equations in two variables be a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0,

If \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\), then

i) the pair of linear equation is dependent and consistent.

ii) the graph will be a pair of coincident lines. Each point on the lines will be a solution and so the pair of equations will have infinitely many solutions.

Therefore, the pair of equations represent a pair of coincident lines.

✦ Try This: Do the equations 4x + y - 1 = 0 and 2x + 9y = 5 represent a pair of coincident lines? Justify your answer.

Given, the pair of equations are

4x + y - 1 = 0

2x + 9y = 5

We have to determine if the equations represent a pair of coincident lines.

Here, a₁ = 4, b₁= 1, c₁ = -1

a₂ = 2, b₂= 9, c₂ = -5

So, a₁/a₂ = 4/2 = 2

b₁/b₂ = 1/9

c₁/c₂ = -1/-5 = 1/5

\(\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}\)

Therefore, the given equations do not represent a pair of coincident lines

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 3

NCERT Exemplar Class 10 Maths Exercise 3.2 Problem 2 (ii)

-2x - 3y = 1; 6y + 4x = -2, do these equations represent a pair of coincident lines

Summary:

The equations -2x - 3y = 1; 6y + 4x = -2 represent a pair of coincident lines.

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