Verified Answer :
- Represent the equation \(4x = -4x\) as two linear functions \(y = 4x\) and \(y = -4x\).
- Graph both functions on the same Cartesian plane to observe their intersection point.
- Identify the point of intersection at \((0,0)\) as the solution, where the \(x\)-coordinate is \(0\).
- The solution to the equation is \(\boxed{x=0}\).
Explanation
- Representing the Equation as a System of Functions
To find the solution(s) to the equation \(4x = -4x\), we can treat it as a system of two linear functions. We equate the left side of the equation to \(y\) and the right side of the equation to \(y\). This gives us two functions:
\[f(x) = 4x\]
\[g(x) = -4x\]
The solutions to the equation \(4x = -4x\) correspond to the \(x\)-coordinates of the points where the graphs of these two functions intersect on the Cartesian coordinate plane. We are essentially solving the system of equations:
\[y = 4x\]
\[y = -4x\]
- Analyzing the Graphs and Finding the Intersection
To graph these, observe that both \(y = 4x\) and \(y = -4x\) are linear equations passing through the origin \((0,0)\) because they have no constant terms.
- The function \(f(x) = 4x\) represents a line with a slope of \(4\) that passes through the origin.
- The function \(g(x) = -4x\) represents a line with a slope of \(-4\) that also passes through the origin.
Since both lines pass through the origin \((0,0)\), the point \((0,0)\) is the only point of intersection. Substituting \(x = 0\) into both original expressions gives \(4(0) = 0\) and \(-4(0) = 0\), which confirms that \(x = 0\) is the solution.
- Final Conclusion
The graph that can be used to find the solution consists of two straight lines on a single coordinate system. Both lines intersect at exactly one point, the origin \((0,0)\). Therefore, the \(x\)-coordinate of the intersection point, which is \(x = 0\), is the solution to the equation \(4x = -4x\). We conclude that the graphical representation of the system \(y = 4x\) and \(y = -4x\) shows the solution clearly at the origin. The final solution is \(\boxed{x=0}\).
Examples
Graphical methods are vital in engineering and economics to find "equilibrium points" where two opposing factors meet. For instance, if you are modeling the profit of a business where \(P(x) = 4x\) represents revenue and \(C(x) = -4x + 10\) represents costs, the point where these graphs intersect is the break-even point. Similarly, here, visualizing the intersection of two lines helps identify the exact input value where two different processes or variables yield the same output.