Problem 72 - Introductory Algebra for College Students

Question

When using the addition method, how can you tell if a system of linear equations has no solution?

Step-by-Step Solution

Answer

In the Addition/Elimination method, if all variables are eliminated and results in an untrue statement after the addition or subtraction process, this indicates that the system of equations has no solution.

Step 1: Understand Addition/Elimination Method

The Addition/Elimination method involves adding or subtracting the equations in order to eliminate one of the variables. This allows you to solve for the remaining variable.

Step 2: Analyze the System of Equations

When using the Addition/Elimination method to solve a system of linear equations, it's possible to end up with an equation that is not possible, such as 0 = 5. This happens when both equations are land up on top of each other after the addition or subtraction process, which is possible when the two lines represented by the equations are parallel.

Step 3: Identify Indication of No Solution

If after adding or subtracting the two equations in the system, you find that all the variables cancel each other out and leave behind an untrue statement (like 0 = 5), this means the system of equations has no solution. This is because you've essentially proved that the two lines never intersect, thus there is no common solution to both equations.