- We represent the equation \(x^2 + 6x = x^2 - 6x\) as the system of equations \(y = x^2 + 6x\) and \(y = x^2 - 6x\).
- Simplifying the original equation \(x^2 + 6x = x^2 - 6x\) leads to \(12x = 0\), which gives the unique solution \(x = 0\).
- Since \(x = 0\) is the only solution, none of the provided options \(-9, -6, 6\) are correct.
- The final solution is \(x = 0\).
Explanation
- Representing the Equation as a System
The objective is to represent the equation \(x^2 + 6x = x^2 - 6x\) as a system of two equations and to find the solutions to the original equation. To express an equation of the form \(f(x) = g(x)\) as a system of equations, we set each side of the equation equal to a common variable, such as \(y\), leading to the system \(y = f(x)\) and \(y = g(x)\). Here, our functions are \(f(x) = x^2 + 6x\) and \(g(x) = x^2 - 6x\). Thus, the corresponding system of equations is:
\[ \begin{cases} y = x^2 + 6x \ y = x^2 - 6x \end{cases} \]
Comparing this to the provided choices, the first option correctly matches our system.
- Solving for x and Verifying Solutions
To find the solutions to the original equation \(x^2 + 6x = x^2 - 6x\), we isolate the variable \(x\):
- Start with the equation: \(x^2 + 6x = x^2 - 6x\).
- Subtract \(x^2\) from both sides of the equation:
\(6x = -6x\).
- Add \(6x\) to both sides:
\(12x = 0\).
- Divide by \(12\):
\(x = 0\).
The only solution to the original equation is \(x = 0\). Examining the given options for the solutions (\(-9, -6, 6\)), we find that none of them satisfy the equation \(x^2 + 6x = x^2 - 6x\). Therefore, none of the provided choices are solutions.
Examples
Solving systems of equations by intersection is a fundamental technique used in various fields like economics, engineering, and physics. For example, in economics, setting two functions representing supply and demand equal to each other allows you to find the equilibrium price and quantity in a market. In this problem, we practiced expressing an equality as a system of functions, which helps visualize solutions as the points where two graphs cross each other on a coordinate plane.