Problem 72 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I think that the nonlinear system consisting of \(x^{2}+y^{2}=36\) and \(y=(x-2)^{2}-3\) is easier to solve graphically than by using the substitution method or the addition method.

Step-by-Step Solution

Answer

The statement makes sense. Solution to the system may be easier graphically due to the complexity involved in solving these equations via substitution or addition methods.

Step 1: Analyze the System of Equations

The first step is to look at the structure of the system of equations. The first equation represents a circle of radius 6 centered at the origin, and the second equation represents a parabola. Because these are both nonlinear equations, neither substitution nor addition methods would be straightforward to apply, because both of them necessitate isolating a variable—something that's difficult to do without resorting to square roots or other complex algebraic manipulations.

Step 2: Consider the Graphical Method

The graphical method in this case would consist of separately graphing the two equations on the same graph, and then identifying the points of intersection, which represent the solutions to the system. While this method does require some proficiency with graphing, it's much more straightforward once you understand how to graph these kinds of equations.

Step 3: Comparison and Conclusion

Given that the addition and substitution methods would require cumbersome manipulations to solve these equations, plus the fact that the graphical method only necessitates basic graphing skills, the statement does make sense—graphing would indeed be an easier method to solve this system of equations.