If a pair of linear equations is consistent, then the lines will be
a. parallel
b. always coincident
c. intersecting or coincident
d. always intersecting

Solution:

We have to determine the nature of lines when a pair of linear equations is consistent.

We know that,

a) 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 equations 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.

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

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

i) the pair of linear equations is consistent

ii) the graph will be a pair of lines intersecting at a unique point, which is the solution of the pair of equations.

From the above results, the pair of linear equations is consistent,

1) If \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\), then the lines will be coincident.

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

Therefore, the lines will be intersecting or coincident.

✦ Try This: If the pair of equations 2x + 3y = 0; 3x + 5y = 0 are consistent, the lines will be

Given, the pair of equations are

2x + 3y = 0

3x + 5y = 0

We have to find the nature of the lines.

Here, a₁ = 2, b₁ = 3

a₂ = 3, b₂ = 5

So, a₁/a₂ = 2/3

b₁/b₂ = 3/5

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

Therefore, the lines will be intersecting at a unique point.

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

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NCERT Exemplar Class 10 Maths Exercise 3.1 Problem 3

If a pair of linear equations is consistent, then the lines will be, a. parallel, b. always coincident, c. intersecting or coincident, d. always intersecting

Summary:

If a pair of linear equations is consistent, then the lines will be intersecting or coincident

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