Algebra

Linear equation in two variables:

An equation of the form, ax + by + c = 0, where a, b and c are constants, such that a and b are both not zero and x and y are variables is called a linear equation in two variables.

For example, 2x + 3y + 10 = 0, 3x + 7y = 0

• When two or more different linear equations in two variables having same variables are taken together, then they form a system of linear equations in two variables.

For example, 2x – 3y = 14 and 2x + y = 6

• To check whether an ordered pair is the solution of a given system of linear equations or not, we just need to verify whether the ordered pair satisfies both the equations or not.

For example, (4, –2) is the solution of 2x – 3y = 14 and 2x + y = 6 as it satisfies both the equations.

• A system of linear equations in two variables is said to form a system of simultaneous linear equations, if each of the equation is satisfied by the same pair of values of the two variables.

Inconsistent, Consistent, and Dependent Pairs of Linear Equations

Let a1x + b1y + c1 = 0, a2x +b2y + c2 = 0 be a system of linear equations.

A pair of linear equations in two variables can be solved by

Graphical method Algebraic methodAlgebraic method to solve linear equations

Case (i)

In this case, the given system is consistent.

This implies that the system has a unique solution.

Case (ii)

In this case, the given system is inconsistent.

This implies that the system has no solution.

Case (iii)

In this case, the given system is dependent and consistent.

This implies that the system has infinitely many solutions.

Example:

Find whether the following pairs of linear equations have unique solutions, no solutions, or infinitely many solutions?

7x + 2y + 8 = 0

14x + 4y + 16 = 0

2x + 3y – 10 =05x – 2y – 6 = 0

3x – 8y + 12 = 06x – 16y + 14 = 0

Solution:

7x + 2y + 8 = 0

14x + 4y + 16 = 0

Here, a1 = 7, b1 = 2,…

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