Problem 93 - Algebra and Trigonometry Real Mathematics, Real People

Question

When solving a system of equations by substitution, how do you recognize that the system has no solution?

Step-by-Step Solution

Answer

If, while applying the substitution method to a system of equations, one derives a false statement, the system is deemed to have no solution. This scenario represents parallel lines that do not intersect.

Step 1- Understand the concept of No Solution

In a system of equations, 'no solution' means that the lines do not intersect, and therefore there is no x, y (or z) that satisfies both equations at the same time. When the systems of equations have the same slope and different y-intercepts, they are parallel and never cross each other. Hence, there will be no solution for such systems.

Step 2 - Implementation of substitution method

Begin by solving one of the equations for either of the variables. Substitute the expression obtained into the other equation of the system. The result is an equation in one variable.

Step 3- Carrying out necessary operations

Solve the single-variable equation. If at this point the result is a true mathematical statement such as 4=4, it confirms the system of equations has an infinite number of solutions. If the resulting statement is false, such as 5=0, this means the system of equations has no solution.