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x + 3y = 11; 2(2x + 6y) = 22, are these pair of linear equations consistent

Solution:

Given, the linear equations are

x + 3y = 11

2(2x + 6y) = 22.

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

The equation 2(2x + 6y) = 22 can be written as 2x + 6y = 11.

Here, a₁ = 1, b₁ = 3, c₁ = 11

a₂ = 2, b₂ = 6, c₂ = 11

So, a₁/a₂ = 1/2

b₁/b₂ = 3/6 = 1/2

c₁/c₂ = 11/11 = 1

1/2 = 1/2 ≠ 1

We know that,

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

If \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}\), then the pair of linear equation is inconsistent

Therefore, the pair of equations is not consistent.

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

2x + 3y = a; 8x + 12y - 4a = 0; a ≠ 0

Given, the equations are

2x + 3y = a

8x + 12y - 4a = 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 = 8, b₂ = 12, c₂ = 4a

So, a₁/a₂ = 2/8 = 1/4

b₁/b₂ = 3/12 = 1/4

c₁/c₂ = a/4a = 1/4

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

Therefore, the pair of equations are consistent

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

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

x + 3y = 11; 2(2x + 6y) = 22, are these pair of linear equations consistent?

Summary:

The pair of linear equations x + 3y = 11; 2(2x + 6y) = 22 is not consistent.

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