(1/2)x + y + (2/5) = 0; 4x + 8y + (5/16) = 0, do these equations represent a pair of coincident lines
Solution:
Given, the pair of equations are
(1/2)x + y + (2/5) = 0
4x + 8y + (5/16) = 0
We have to determine if the pair of equations represent a pair of coincident lines.
Here, a1 = 1/2, b1 = 1, c1 = 2/5
a2 = 4, b2 = 8, c2 = 5/16
So, a1/a2 = (1/2)/4 = 1/8
b1/b2 = 1/8
c1/c2 = (2/5)/(5/16) = 32/25
So, 1/8 = 1/8 ≠ 32/25
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 graph will be a pair of parallel lines.
Therefore, the given pair of equations have no solution.
✦ Try This: Do the equations 3x - y + 8 = 0 and 6x - 2y = -16 represent coincident lines?
Given, the pair of equations are
3x - y + 8 = 0
6x - 2y = -16
We have to determine if the pair of equations represent coincident lines.
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.
Here, a1 = 3, b1 = -1, c1 = 8
a2 = 6, b2 = -2, c2 = 16
So, a1/a2 = 3/6 = 1/2
b1/b2 = -1/-2 = 1/2
c1/c2 = 8/16 = 1/2
So, 1/2 = 1/2 = 1/2
\(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
Therefore, the pair of equations represent coincident lines
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 3
NCERT Exemplar Class 10 Maths Exercise 3.2 Problem 2 (iii)
(1/2)x + y + (2/5) = 0; 4x + 8y + (5/16) = 0, do these equations represent a pair of coincident lines
Summary:
The pair of equations (1/2)x + y + (2/5) = 0; 4x + 8y + (5/16) = 0 does not represent a pair of coincident lines
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