Problem 52 - Precalculus with Limits

Question

Choosing a Solution Method In Exercises \(49-56\) , solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{l}{y=x^{3}-2 x^{2}+x-1} \\ {y=-x^{2}+3 x-1}\end{array}\right.$$

Step-by-Step Solution

Answer

The solutions to the system are at the intersection points of the curves represented by the equations when graphed. Validation of these solutions can be done by substituting back into the original equations.

Step 1: Selection of the Solution Method

Observe the given system. Both equations are non-linear. Solving this system algebraically could be quite complex due to the raised powers of x in the equations. It's easier to solve this system graphically. We will plot both equations on a graph and find the intersection points.

Step 2: Graphing the Equations

Plot both curves \(y = x^{3}-2x^{2}+x-1\) and \(y = -x^{2}+3x-1\). This can be done using a graphing calculator or mathematical software. Remember, the points where both of these curves intersect are the solutions to the system.

Step 3: Finding the Solutions

Locate the points at which the curves intersect on your graph. These coordinates are your solutions to the system of equations. There could be one, many, or no solutions, depending on the equations and where they intersect.

Step 4: Validation of the Solutions

To validate the solutions, you can substitute the coordinates of the intersection points into both of the original equations. If both equations are true with these values, then these are the solutions of the system.

Key Concepts

Graphical Solution Method

When tackling a system of equations, one effective way of finding the solution is by using the graphical method. This involves plotting the equations on a graph and visually identifying the points where the graphs intersect. For the given system of equations:

• y = x^3 - 2x^2 + x - 1
• y = -x^2 + 3x - 1

we observe that both equations involve non-linear terms. The presence of powers higher than one suggests that algebraic manipulation might be cumbersome. Here, the graphical method becomes advantageous as it provides a visual interpretation of the solution.
Using a graphing calculator or software, we can plot these equations. The curves will intersect at certain points on the plane, representing solutions to the system. These intersection points are where both equations have the same x and y values.
It's essential to ensure accurate plotting for precise results. By checking for clear intersections, we can determine if there are multiple solutions or if none exist.

Algebraic Solution Method

The algebraic solution method involves solving each equation analytically to find common solutions. For linear equations, methods like substitution or elimination are straightforward and effective. However, with non-linear systems, as in this exercise, the algebraic approach can become complex and require higher-level algebra skills.
In the given system, both equations feature terms with variables raised to powers of two and three. Solving such systems algebraically often involves setting the equations equal to each other or isolating one variable and substituting it into the other equation. However, the polynomial nature with mixed degree terms complicates the process, possibly leading to intricate calculations.

• This method might require factoring or applying the quadratic formula in certain scenarios.
• It's also important to consider checking for extraneous solutions, as they might appear during manipulation of non-linear equations.

Ultimately, while possible, solving non-linear systems algebraically can be daunting without technological assistance or advanced techniques, which is why graphical methods are sometimes preferred.

Intersection of Functions

The intersection of functions refers to the points at which two graphs meet on a plane. These points represent the solutions to a system of equations because they satisfy both equations simultaneously. For the system specified in the exercise, identifying intersections provides a clear answer without the need for complex calculations.
Visually analyzing the curves:

• The first function, y = x^3 - 2x^2 + x - 1, produces a cubic curve.
• The second, y = -x^2 + 3x - 1, forms a downward-opening parabola.

Plotting these, we look for their meeting points. Each intersection implies a solution where both y-values are equal at the same x-value, satisfying both equations. When confirming these solutions, substituting back the intersection coordinates into the original equations will verify their accuracy.
The beauty of finding intersections graphically is the instantaneous insight it provides into the behavior and solutions of functions, simplifying otherwise intricate systems.