- Identify the equation as a pair of functions \(f(x) = x^2 + 6x\) and \(g(x) = x^2 - 6x\).
- Translate the equality \(f(x) = g(x)\) into a system of two equations by setting \(y\) equal to each side: \(y = f(x)\) and \(y = g(x)\).
- Recognize that the resulting system \(y = x^2 + 6x\) and \(y = x^2 - 6x\) matches choice (c).
- The final answer is \(\boxed{\text{c}}\).
Explanation
- Understanding the Representation of an Equation as a System
To represent an equation of the form \(f(x) = g(x)\) as a system of equations, we look for two equations that involve a variable \(y\), such that substituting \(y\) leads back to the original equation. We can set \(y\) equal to each side of the equation:
\[y = f(x)\]
\[y = g(x)\]
By setting these two equations equal to each other (\(f(x) = g(x)\)), we find the points of intersection for the two graphs, which corresponds exactly to the solutions for \(x\) in the original equation.
- Applying the Concept to the Given Equation
Given the equation \(x^2 + 6x = x^2 - 6x\), we can identify the two functions on each side:
\[f(x) = x^2 + 6x\]
\[g(x) = x^2 - 6x\]
To create a system of equations that reflects this equality, we set \(y\) equal to each expression:
\[y = x^2 + 6x\]
\[y = x^2 - 6x\]
This system represents the intersection of the two parabolas \(y = x^2 + 6x\) and \(y = x^2 - 6x\). Any point \((x, y)\) that satisfies both equations must have the same \(y\)-coordinate, meaning \(x^2 + 6x = x^2 - 6x\).
- Selecting the Correct System
We compare our derived system:
\[\begin{array}{l} y = x^2 + 6x \ y = x^2 - 6x \end{array}\]
with the provided choices:
A\(\begin{array}{l} y = -x^2 - 6x \ y = x^2 - 6x \end{array}\)
B\(\begin{array}{l} y = -x^2 + 6x \ y = x^2 - 6x \end{array}\)
C\(\begin{array}{l} y = x^2 + 6x \ y = x^2 - 6x \end{array}\)
Choice (c) matches our derived system exactly. Therefore, the correct representation of the equation as a system is choice (c).
Examples
In data science and economics, finding the equilibrium point between supply and demand is a classic application of this method. For example, if a demand function is \(D(p) = 100 - 2p^2\) and a supply function is \(S(p) = p^2 + 10\), the equilibrium price occurs where \(D(p) = S(p)\), which can be visualized as the intersection of the systems \(y = 100 - 2p^2\) and \(y = p^2 + 10\). Solving this system helps businesses and researchers determine the optimal price at which supply matches demand.