Problem 73 - College Algebra

Question

When using the addition or substitution method, how can you tell if a system of linear equations has infinitely many solutions? What is the relationship between the graphs of the two equations?

Step-by-Step Solution

Answer

A system of linear equations has infinitely many solutions when the two equations are equivalent and thus form the same line when graphed. Algebraically, this happens when one equation is a scalar multiple of another. This case graphically corresponds to the two equations representing the same line, meaning they overlap entirely. So every point on the line is a solution for both of the equations, yielding infinite solutions.

Step 1: Understand the System of Linear Equations

The system of linear equations is a collection of two or more equations with the same set of unknown variables. We are looking at a system with two equations in this exercise.

Step 2: Explore Addition or Substitution Method

The addition or substitution method is used to find the solutions of a system of linear equations. By these methods, we look for a common piece of information between the equations, so that one equation can either be added to or replaced by another equation.

Step 3: Identify the Condition for Infinitely Many Solutions

A system of linear equations has infinitely many solutions when the two equations are equivalent, meaning they form the same line when graphed. In algebraic terms, this happens when one equation is a scalar multiple of the other. In other words, if you can multiply all the terms in one equation by the same number to get the other equation, then they are equivalent equations and the system has infinitely many solutions.

Step 4: Understand the Graphical Interpretation

In terms of graphs, if two equations are representing the same line, they will coincide with each other entirely. This means that every point on the line is a solution for both of the equations. That's why there are an infinite number of (x, y) pairs or solutions.