Problem 70 - Intermediate Algebra

Question

When using the elimination (addition) method, how can you tell whether a. a system of linear equations has no solution? b. a system of linear equations has infinitely many solutions?

Step-by-Step Solution

Answer

No solution: parallel lines, result in contradiction; infinitely many solutions: result is a true identity, lines overlap.

Step 1: Understanding the Elimination Method

In the elimination method, we manipulate the given linear equations to eliminate one of the variables by adding or subtracting the equations, aiming to solve for the other variable(s). The result can help us determine if there is a single solution, no solution, or infinitely many solutions.

Step 2: Identifying No Solution

A system of linear equations has no solution when, after using the elimination method, the result is a false statement such as \(0 = c\) where \(c\) is a non-zero constant. This indicates that the lines are parallel and do not intersect.

Step 3: Identifying Infinitely Many Solutions

A system of linear equations has infinitely many solutions if the result of the elimination process is a true statement like \(0 = 0\). This indicates that the equations are dependent and the lines overlap completely, representing the same line.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations that share the same set of variables. The goal when solving such a system is to find all possible values for the variables that satisfy every equation in the system. Linear equations, which graph as straight lines on a coordinate plane, can have an intersection point, overlap entirely, or never meet.
To solve these systems, several methods are available, including substitution, elimination, and graphing. The elimination method, often referred to as the addition method, involves adding or subtracting equations to remove one variable at a time. By reducing variables, this method can help solve the equations for the other variables. It’s an efficient way to determine if the system has a single solution, no solution, or infinitely many solutions. Each scenario provides insights into the relationship between the equations in the system.
Using the elimination method effectively requires practice. It's especially suitable when the equations are already in a format that makes variable cancellation straightforward. Identifying how many solutions exist will depend on the result after using the elimination strategy.

No Solution

A system of linear equations has no solution when the elimination method results in a contradiction, which means we get a statement that is impossible. Specifically, if you end up with an equation like \(0 = c\) where \(c\) is a non-zero constant, this indicates there's no solution. This happens because the lines represented by the equations are parallel and never intersect.
Parallel lines have the same slope but different y-intercepts, meaning they will always remain the same distance apart and will not meet. Such systems are called inconsistent.

• Example: Consider the equations \(2x + 3y = 6\) and \(2x + 3y = 12\). If you subtract these equations, you get \(0 = 6\), which is a contradiction.

When using the elimination method, always check the resulting statement. A contradictory result like \(0 = 6\) means the system has no possible solutions.

Infinitely Many Solutions

Infinitely many solutions occur when the system of linear equations is essentially the same line or multiple lines on top of each other. When you use the elimination method and end up with a statement like \(0 = 0\), this means every part of both equations matches, showing that the lines are coincident.
Coincident lines overlap completely because they have the same slopes and y-intercepts, resulting in infinitely many intersection points. This scenario is often referred to as a dependent system.

• Example: Take the equations \(x - y = 1\) and \(2x - 2y = 2\). By multiplying the first equation by 2, it becomes the second equation.

Thus, when solving using elimination, if you arrive at a real identity such as \(0 = 0\), it reveals that there are infinitely many solutions. This is a clear indication that both equations represent the same line in the plane.