Problem 90 - College Algebra Essentials

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 represent the same line (are coincident). This occurs when one equation is a multiple of the other, resulting in an identity like \(0=0\) when using the addition or subtraction methods. This alines with the idea that the graphs of the two equations will coincide for there to be infinitely many solutions. The ratio of coefficients and constants will be equal.

Step 1: Understanding the Addition or Subtraction Method

In the addition or subtraction method, you add or subtract the given equations in a way to eliminate one of the variables and then solve for the remaining variable. The aim is to simplify the system of linear equations thus making it easier to solve.

Step 2: Identifying Infinitely Many Solutions

A system of linear equations has infinitely many solutions if after the addition or subtraction step, we get an identity, such as \(0=0\) or \(5=5\). This results in a situation where both the x and y variables are eliminated, leaving only a true statement, which means the system has infinitely many solutions.

Step 3: Understanding the Relationship between the Graphs

If a system of linear equations has infinitely many solutions, the graphs of the equations will coincide, i.e., they will be the same line. In other words, every point that lies on one line also lies on the other.

Step 4: Describing the Relationship of the Coefficients

In a system of linear equations, if the ratio of the coefficients of one variable is equal to the ratio of the constants, then the equations have infinitely many solutions. In the equations \(ax+by=c\) and \(dx+ey=f\), if \(a/d = b/e = c/f\), then the system of the given equations has infinitely many solutions.