- Rewrite the given system as a vertical parabola \(y = x^2 + 5x - 10\) and a line \(y = -5x + 1\).
- Substitute the line equation into the parabola to obtain the quadratic equation \(x^2 + 10x - 11 = 0\).
- Calculate the discriminant \(D = 10^2 - 4(1)(-11) = 144\).
- Since \(D > 0\), the quadratic equation has two distinct real solutions for \(x\), and thus the system has \(\boxed{2}\) solutions.
Explanation
- Analyzing the Equations
We are given a system of two equations with variables \(x\) and \(y\):
1) \(10 + y = 5x + x^2\)
2) \(5x + y = 1\)
First, let us analyze the nature of these equations. The first equation, \(10 + y = 5x + x^2\), can be rearranged as \(y = x^2 + 5x - 10\). This is the standard form of a vertical parabola. The second equation, \(5x + y = 1\), can be rewritten as \(y = -5x + 1\), which represents a straight line. The problem asks for the number of solutions \((x, y)\) that satisfy both equations simultaneously.
- Setting up the Quadratic Equation
To find the intersection points (solutions) of the parabola and the line, we substitute the expression for \(y\) from the second equation into the first equation:
\[ -5x + 1 = x^2 + 5x - 10 \]
Rearranging this to form a standard quadratic equation of the form \(ax^2 + bx + c = 0\):
\[ x^2 + 5x + 5x - 10 - 1 = 0 \]
\[ x^2 + 10x - 11 = 0 \]
- Calculating the Discriminant
We have the quadratic equation \(x^2 + 10x - 11 = 0\). To determine the number of real solutions for \(x\), we calculate the discriminant \(D = b^2 - 4ac\), where \(a = 1\), \(b = 10\), and \(c = -11\):
\[ D = (10)^2 - 4(1)(-11) \]
\[ D = 100 + 44 \]
\[ D = 144 \]
- Conclusion on Number of Solutions
The discriminant \(D = 144\) is greater than 0. This implies that the quadratic equation \(x^2 + 10x - 11 = 0\) has two distinct real roots for \(x\). Since every distinct real root \(x\) corresponds to exactly one value of \(y\) via the linear equation \(y = -5x + 1\), there are two distinct pairs of \((x, y)\) that satisfy the system. Therefore, the system of equations has 2 solutions.
Examples
In architecture and engineering, finding intersection points of different structural components is crucial for stability. For instance, if you are designing a suspension bridge, the cables often follow parabolic paths, while other supports may be straight. Solving systems like this helps engineers calculate exactly where these parts meet, ensuring they connect accurately to support the bridge structure safely and effectively.