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3x + (1/7)y = 3; 7x + 3y = 7, do these equations represent a pair of coincident lines?

Solution:

Given, the pair of equations are

3x + (1/7)y = 3

7x + 3y = 7

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

Here, a₁ = 3, b₁= 1/7, c₁ = -3

a₂ = 7, b₂ = 3, c₂ = -7

So, a₁/a₂ = 3/7

b₁/b₂ = (1/7)/3 = 1/21

c₁/c₂ = -3/-7 = 3/7

3/7 ≠ 1/21

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

We know that,

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

If \(\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\), then the graph will be a pair of lines intersecting at a unique point, which is the solution of the pair of equations.

Therefore, the pair of equations has a unique solution.

✦ Try This: The pair of equations 2x - 16y = 8 and 4x - 28y = 16 has

Given the pair of equations are

2x - 14y = 8

4x - 28y = 16

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

Here, a₁ = 2, b₁ = -14, c₁ = -8

a₂ = 4, b₂ = -28, c₂ = -16

So, a₁/a₂ = 2/4 = 1/2

b₁/b₂ = -14/-28 = 1/2

c₁/c₂ = 8/16 = 1/2

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

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 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 given pair of equations has infinitely many solutions

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

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

3x + (1/7)y = 3; 7x + 3y = 7, do these equations represent a pair of coincident lines

Summary:

The pair of equations 3x + (1/7)y = 3; 7x + 3y = 7 does not represent a pair of coincident lines.

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