Problem 90 - Algebra and Trigonometry

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

If the addition or substitution method results in a true statement like 0=0, the system of linear equations has infinitely many solutions. Graphically, the lines representing the equations in the system coincide entirely, meaning they have the same gradient and y-intercept.

Step 1: Recognizing the System of Equations

When working with a system of equations, solutions can be considered as points where the equations intersect when graphed. So if there are infinitely many solutions, it means that the equations coincide and therefore the graphs overlap completely.

Step 2: Checking Characteristic in Addition or Substitution Method

In both the addition and substitution methods, if it leads to a true statement, such as 0 = 0, it means the original system actually represents the same linear equation and has infinitely many solutions.

Step 3: Considering the graphical representation

In a graphical context, if a system has infinitely many solutions, the lines representing the equations will be the exact same line i.e., they coincide - every point on the line is an intersection. This means that the equations are just re-arrangements or scalings of each other, thus having the same gradient and y-intercept, leading to infinitely many points of intersection.