Exercise 21 Page 411

A system of linear equations with two linear equations can have zero, one or infinitely many solutions. It is possible to find the number of solutions of a system by solving the equations algebraically. To do so, we need to check the result we get by using either the Substitution Method or the Elimination Method.

1. One solution: The result is a solution for one variable.
2. No solution: The result is a false statement.
3. Infinitely many solutions: The result is an identity.

We are told that we got 0=0 after adding two linear equations in a system. This is a true statement no matter the value of x and y and is called an identity. Hence, any point is a solution to such a system. In other words, the system of equations will have infinitely many solutions.

Extra

Example

To make sense of the explanation, let's consider an example system that will result in 0=0 when we add the equations in the system. 2x-y= 2 & (I) y-2x= - 2 & (II) We will add these equations. 2x-y+ y-2x= 2+( - 2) ⇓ 0=0

We arrived at an identity. It means that the statement is always true regardless of the values of x and y. Therefore, the lines that represent the equations are coincident, and there are infinitely many solutions to the system. Let's look at the graph of the system to visualize it.