x = 2y; y = 2x, does this pair of linear equations have no solution
Solution:
Given, the pair of linear equations is
x = 2y
y = 2x
We have to determine whether the pair of equations has a solution or not.
The equations can be rewritten as
x - 2y = 0
2x - y = 0
Here, a₁ = 1, b₁ = -2, c₁ = 0
a₂ = 2, b₂ = -1, c₂ = 0
So, a₁/a₂ = 1/2
b₁/b₂ = -2/-1 = 2
c₁/c₂ = 0
1/2 ≠ 2
\(\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\)
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: Do the pair of linear equations 2x + 4y = 3; 6y + 4x = 9 have no solution? Justify your answer.
Given, the pair of equations are
2x + 4y = 3
6y + 4x = 9
We have to determine whether the pair of equations has a solution or not.
The equation 6y + 4x = 9 can be rewritten as 4x + 6y = 9.
Here, a₁ = 2, b₁ = 4, c₁ = 3
a₂ = 4, b₂ = 6, c₂ = 9
So, a₁/a₂ = 2/4 = 1/2
b₁/b₂ = 4/6 = 2/3
c₁/c₂ = 3/9 = 1/3
1/2 ≠ 2/3 ≠ 1/3
\(\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\)
Therefore, the pair of equations has a unique solution
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 3
NCERT Exemplar Class 10 Maths Exercise 3.2 Problem 1 (ii)
x = 2y; y = 2x, does this pair of linear equations have no solution
Summary:
The pair of linear equations x = 2y; y = 2x has a unique solution.
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