Problem 23 - College Algebra and Calculus: An Applied Approach

Question

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{aligned} 6 r-5 s &=3 \\\\-1.2 r+s &=0.5 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The given system of linear equations is inconsistent and has no solution.

Step 1: Standardize the system

Rewrite the system so that equation coefficients can easily be eliminated and handled. Let's multiply the second equation by \(5\) to facilitate further elimination. This gives us the system:\[\begin{aligned}6r - 5s &= 3, \-6r + 5s &= 2.5.\end{aligned}\]

Step 2: Eliminate one variable

The coefficients of \(r\) and \(s\) in both equations are now the same (but with opposite signs). We can add the two equations to eliminate these variables. Adding them gives:\[0 = 5.5.\]

Step 3: Determine consistency

As the simplified equation does not make mathematical sense (no nonzero number can equal zero), we can conclude that our original system of equations has no solution. Therefore, the given system is inconsistent.

Key Concepts

Elimination Method

The elimination method is a popular technique for solving systems of linear equations. It relies on removing or eliminating one of the variables to simplify the equations into something solvable.
Here’s a quick guide on how we apply this method:

• First, choose which variable you want to eliminate. Generally, we look at the coefficients and decide based on which one seems easiest to work with.
• Then, adjust the equations by multiplying one or both of them by suitable numbers. The goal is to ensure that when you add or subtract the equations, the chosen variable cancels out.
• Once a variable is eliminated, you'll get a simpler equation with just one variable. Solve this equation.
• Finally, substitute the found value back into one of the original equations to solve for the other variable.

Inconsistent systems, as you'll see, might not yield a solution after elimination.

Consistent Systems

Consistent systems are systems of equations that have at least one solution. Solutions can be either a single point where the lines intersect or infinitely many solutions if the lines overlap entirely.
Here's what you might need to identify a consistent system:

• When solving, if you reach a statement like "0 = 0," it usually means there are infinitely many solutions and the system is dependent, meaning the equations represent the same line.
• If instead you find a particular value for each variable, you have a single intersection point or solution.

In our step-by-step solution, we concluded that this particular system is not consistent because we reached an impossibility (0 = 5.5).

Inconsistent Systems

An inconsistent system of equations is a system that has no solution. This happens when the equations represent parallel lines, which do not intersect ever.
When working through an elimination method, here’s how to spot an inconsistency:

• After simplifying the system through elimination, you might end up with a logically false statement like "0 = 5.5." This indicates that the terms on left and right don't match up mathematically, so no solution exists.
• This means the lines are parallel in a geometric sense. Visualizing this can help students understand why there's no solution.

An inconsistent system tells us that regardless of the values assigned to the variables, the equations can’t be true simultaneously.