Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}6.3 x+7.2 y=5.4 \\ 5.6 x+6.4 y=4.8\end{array}\right.\)

Step-by-Step Solution

Answer

The system of equations has infinite solutions, as the two equations are dependent.

Step 1: Equalise Coefficients

Using the method of elimination, either the \(x\)s or \(y\)s need to have the same coefficient. We can do this by multiplying all terms of the first equation by \(5.6\) and those of the second equation by \(6.3\). This gives us : \[\left\{\begin{array}{l}35.28x+40.32y=30.24 \ 35.28x+40.32y=30.24 \end{array}\right.\]

Step 2: Subtract the equations

Now, since the left-hand sides of the two equations are the same, we can subtract the second equation from the first. The terms with the variable \(x\) cancel out, giving us a resulting equation of: \(0=0\). This implies that the original two equations are dependent and have infinite solutions

Step 3: Check our solution

To confirm that there are indeed infinite solutions, a graphing utility can be used. By graphing the two equations, it will be seen that they are the same line, confirming that there are infinite solutions. Since we have infinite solutions, we can't quote a specific pair \((x, y)\) as solution.

Key Concepts

Elimination Method

The elimination method is a systematic approach used to solve systems of equations. This method involves adding or subtracting equations so that one of the variables is eliminated, making it possible to solve for the other variable. Here's a simple way to understand this method:

• First, choose which variable you want to eliminate. This could be either the variable "x" or "y" in the given system of equations.
• Next, you need to manipulate the equations so that the coefficients (the numbers in front of the variable) of one variable are equal in both equations.
• This is typically achieved by scaling the equations, meaning you might multiply all terms of one or both equations by a specific number.
• Once the chosen variable has the same coefficient, you can subtract the equations. This will cancel out the chosen variable, leaving an equation with just one variable that can be solved.

This method is particularly useful when the coefficients make it easy to align one of the variables, allowing for a straightforward elimination process.

Dependent Systems

A dependent system is a system of equations where all equations essentially represent the same geometric entity -- typically a line in a two-dimensional space. When you use the elimination method and end up with an equation that is always true, such as "0 = 0", it signifies that the system is dependent. Here's how it works:

• If you scale and combine the equations and they reduce to something that holds for any value, the two original equations describe the same line.
• This means that every solution to one equation is also a solution to the other.
• In terms of graphing, if two lines coincide, they have infinitely many points in common, thus creating a "dependent system."

Dependent systems can be tricky because they do not have a unique solution. Instead, they suggest that the solutions are not independent from each other.

Infinite Solutions

Infinite solutions occur in a system of equations when there is not a unique solution or no solution set that solves the system at distinct points. This typically happens in two specific situations with systems of equations:

• When you have a dependent system, as mentioned before, every point on the line (which can be described by the equation) is a solution.
• This means there's an infinite number of "(x, y)" pairs that satisfy both equations.
• Graphically, if you try to plot a dependent system, you'd find that both equations graph to the same line, reinforcing that all points on the line solve the system.
• An infinite number of solutions indicate that the system is consistent, but it does not provide clear distinct solutions due to overlapping.

This scenario often requires checking the setup of the problem or using additional constraints if possible, to find practical solutions that meet specific conditions within real-world applications.