A linear equation in two variables is an equation of the form ax + by + c = 0 where a, b, c ∈ R, a, and b ≠ 0. A system of linear equations that has no solution is called an inconsistent pair of linear equations. When we consider a system of linear equations, we can compare the coefficients of the equations and find whether it is a system of equations with no solution. We can determine the same from a graph also.
Inconsistent Pair of Linear Equations
| Consider the pair of linear equations in two variables x and y.
a1x + b1y + c1 = 0 a2x + b2y + c2 = 0 Here, a1, b1, c1, a2, b2, c2 are all real numbers. Note that, a12 + b12 ≠ 0, a22 + b22 ≠ 0 If (a1/a2) = (b1/b2) ≠ (c1/c2), then there will be no solution. |
This type of system of equations is called an inconsistent pair of linear equations. If we plot the graph, the lines will be parallel, and the system of equations will have no solution.
Example
Find the value of x and y
-4x + 10y = 6
2x – 5y = 3
Solution:
Given -4x + 10y = 6 …(i)
2x – 5y = 3 …(ii)
Divide (i) by (ii), and reduce it.
We get -2x + 5y = 3
Solve for x
x = (5/2)y – (3/2)
Substitute x in (ii)
2[(5/2)y – (3/2)] – 5y = 3
5y – 3 – 5y = 3
0 = 3 + 3
We get, 0 = 6. This cannot be true.
Hence, the system of equations has no solution.
Frequently Asked Questions
What do you mean by linear equation?
An equation whose variables have a degree 1 is called a linear equation.
Give the condition that the pair of linear equations in 2 variables have no solution.
Suppose a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.
It represents the pair of linear equations in two variables, x and y. If (a1/a2) = (b1/b2) ≠ (c1/c2), then there will be no solution.
How is the graph of a system of linear equations having no solution?
The lines will be parallel if we plot the graph of the system of linear equations having no solution.
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