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System of Linear Equations has No Solution

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. From the graph also, we can determine the same.

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 system of equations 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 2 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 degree 1 is called a linear equation.

Give the condition that the pair of linear equations in 2 variables has no solution.

Suppose a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
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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