Question

How can you tell when a system of three linear equations with three variables has no solution? Infinitely many solutions?

Short answer

While solving whenever we end up with a statement that has no variables and is mathematically false, we know there are no solutions and that the system is inconsistent.

While solving whenever we end up with a statement that has no variables and is mathematically true, we know there are infinitely many solutions. The system is consistent with dependent equations.

Step-by-step solution

Step 1. Definition

A linear equation with three variables, where $a,b,c$, and $d$are real numbers and $a,b$, and $c$are not all zero, is of the form $ax+by+cz=d$. Every solution to the equation is an ordered triple, $(x,y,z)$that makes the equation true.

A system of linear equations with three variables can have no solution, one solution, or infinitely many solutions.

Step 2. System of linear equations having no solution

When we solve a system of three linear equations with three variables we sometimes end up with a statement that contains no variables and is also mathematically a false statement,.

So after seeing that we know there are no solutions and that the system is inconsistent.

Step 3. System of linear equations having infinite solutions

When we solve a system of three linear equations with three variables we sometimes end up with a statement with no variables and is a mathematically true statement.

As the statement is true and has no variables so it does not depends on any variable and thus we know there are infinitely many solutions. The system is consistent with dependent equations