Problem 26 - Precalculus: Mathematical for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{aligned} 25 x-75 y &=100 \\ -10 x+30 y &=-40 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions: \((3y + 4, y)\).

Step 1: Analyze the System of Equations

The given system of equations is:\[\begin{align*}25x - 75y &= 100 \-10x + 30y &= -40\end{align*}\] Both equations appear to be linear and could potentially have a unique solution, infinite solutions, or no solution.

Step 2: Simplify Each Equation

Divide the first equation by 25 and the second equation by -10 to make them easier to analyze. This gives:\[\begin{align*}x - 3y &= 4 \x - 3y &= 4\end{align*}\] The simplified equations are exactly the same.

Step 3: Check for Consistency of the System

Since the simplified equations are identical, they are dependent. This means the system has infinitely many solutions, as any pair \((x, y)\) satisfying this equation is a solution. This indicates that the two lines represented by these equations are coincident.

Step 4: Express the Solution in Ordered-Pair Form

Since the system has infinitely many solutions, express the solution in the form \((x, y)\). Using the equation \(x - 3y = 4\), solve for \(x\) in terms of \(y\): \[x = 3y + 4\] Therefore, the solution in ordered-pair form is \((3y + 4, y)\), where \(y\) is any real number.

Key Concepts

Infinitely Many Solutions

When we talk about a system of linear equations having infinitely many solutions, it means that there are countless combinations of variables that satisfy all the equations at the same time. In the context of our given problem, we had two initially different looking equations that simplify to the same equation.

This indicates that both equations represent the same line on the graph. Therefore, every point on that line is a solution to the system. This happens when the lines overlap completely, meaning they are coincident.

Hence, instead of there being just one meeting point or no meeting points at all, there are innumerable points on that line, translating to infinitely many solutions.

Consistent System

A consistent system of equations is one that has at least one solution. In the cases of linear systems, this means the lines intersect at least once.

Our exercise revealed that the system is consistent because it has infinitely many solutions, indicating all points along the overlapping lines solve both equations. The consistent system further assures us that there’s no contradiction between the equations, allowing for these real solutions.

In contrast, an inconsistent system would arise if there were parallel lines which never meet each other, yielding no solutions at all.

Linear Equations

Linear equations form the backbone of solving systems such as the one given in our exercise. A linear equation involves variables to the first power and graphed, it represents a straight line.

In our exercise, we initially started with:

• 25x - 75y = 100
• -10x + 30y = -40

Though these might look different initially, simplifying them gives us identical linear equations reflecting the same line. In essence:

• Both equations boil down to x - 3y = 4 after simplification.

Hence, understanding linear equations and their simplification techniques can often reveal important solutions or characteristics of a system.

Ordered Pairs

Ordered pairs represent solutions to equations in a very clear-cut way, using the format (x, y). This makes it easy to visualize and apply them in many contexts.

In the exercise, after recognizing there were infinitely many solutions, we expressed them as ordered pairs in the form (3y + 4, y).

• This representation gives y a free rein; it's any real number.
• Consequently, x changes accordingly based on each y you choose.

Ordered pairs are widely used in graphing systems of equations, ensuring clarity on which points satisfy given equations.