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2ax + by = a; 4ax + 2by - 2a = 0; a, b ≠ 0, are these pair of linear equations consistent

Solution:

Given, the equations are

2ax + by = a

4ax + 2by - 2a = 0.

We have to determine if the pair of equations is consistent.

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 pair of equations is consistent.

Here, a₁ = 2a, b₁ = b, c₁ = a

a₂ = 4a, b₂ = 2b, c₂ = 2a

So, a₁/a₂ = 2a/4a = 1/2

b₁/b₂ = b/2b = 1/2

c₁/c₂ = a/2a = 1/2

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

Therefore, the pair of equations is consistent.

✦ Try This: Are the following pair of linear equations consistent? Justify your answer.

2x + 3y = a; 4x + 6y - 2a = 0; a ≠ 0

Given, the equations are

2x + 3y = a

4x + 6y - 2a = 0.

We have to determine if the pair of equations is consistent.

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 pair of equations is consistent.

Here, a₁ = 2, b₁= 3, c₁ = a

a₂ = 4, b₂= 6, c₂ = 2a

So, a₁/a₂ = 2/4 = 1/2

b₁/b₂ = 3/6 = 1/2

c₁/c₂ = a/2a = 1/2

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

Therefore, the pair of equations is consistent

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

NCERT Exemplar Class 10 Maths Exercise 3.2 Problem 3 (iii)

2ax + by = a; 4ax + 2by - 2a = 0; a, b ≠ 0, are these pairs of linear equations consistent

Summary:

The pair of linear equations 2ax + by = a; 4ax + 2by - 2a = 0; a, b ≠ 0 is consistent.

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