Problem 91 - College Algebra

Question

When using the addition or substitution method, how can you tell if a system of linear equations has no solution? What is the relationship between the graphs of the two equations?

Step-by-Step Solution

Answer

For a system of linear equations, if the addition or substitution method leads to a false statement, then the system has no solution. Graphically, such a system is represented by two parallel lines which do not intersect.

Step 1: Identify The Statement Resulting from the Methods

In both the addition and substitution method for solving a system of linear equations, you need to manipulate the equations until you obtain a new equation or a definitive statement. If this process leads to a false statement such as 0=1, it suggests that the system of equations has no solutions.

Step 2: Understand Graphical Interpretation

When a system of linear equations has no solution, their respective graphs are parallel lines. Two parallel lines never intersect hence they do not have a common solution.

Step 3: Linkage Between Step 1 and Step 2

To tie it all together, when handling a system of linear equations, a false statement resulting from substitution or addition method indicates a system with no solution. Graphically, such a system is represented by parallel lines that do not intersect.