Possible Solutions For A System Of Equations Parabola And Line

In mathematics, systems of equations often represent the intersection of different geometric shapes. When dealing with a system comprising a parabola and a line, understanding the possible numbers of solutions becomes a fascinating exploration. This article delves into the system of equations:

$$\begin{array}{l} 10+y=5 x+x^2 \\ 5 x+y=1 \end{array}$$

We aim to determine the possible number of solutions, which geometrically corresponds to the number of intersection points between the parabola and the line. By analyzing the equations and their graphical representations, we will uncover the different scenarios and the conditions that lead to them.

Before diving into the solutions, let's first understand the nature of the given equations. The first equation, 10 + y = 5x + x², represents a parabola. To recognize this, we can rearrange the equation into the standard form of a parabola, which is y = ax² + bx + c. By rearranging the terms, we get:

y = x² + 5x - 10

This equation represents a parabola opening upwards (since the coefficient of x² is positive) with its vertex located at a specific point. The vertex's location and the parabola's shape will influence how it intersects with the line.

The second equation, 5x + y = 1, represents a line. We can rewrite this equation in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Rearranging the terms, we get:

y = -5x + 1

This equation represents a line with a slope of -5 and a y-intercept of 1. The slope and y-intercept determine the line's orientation and position in the coordinate plane, which are crucial factors in determining the intersection points with the parabola.

A powerful way to understand the possible solutions is to visualize the equations graphically. The solutions to the system of equations correspond to the points where the parabola and the line intersect on the coordinate plane. There are three possible scenarios:

1. No Intersection: The line and the parabola do not intersect at any point. In this case, the system of equations has no solutions. Graphically, this means the line passes either above or below the parabola without touching it.
2. One Intersection: The line touches the parabola at exactly one point. This occurs when the line is tangent to the parabola. In this case, the system of equations has one solution, which corresponds to the point of tangency.
3. Two Intersections: The line intersects the parabola at two distinct points. In this case, the system of equations has two solutions, corresponding to the coordinates of the two intersection points.

To determine the exact number of solutions, we can solve the system of equations algebraically. This involves finding the values of x and y that satisfy both equations simultaneously. We can use the substitution method or the elimination method.

Let's use the substitution method. From the second equation, we have y = -5x + 1. We can substitute this expression for y into the first equation:

10 + (-5x + 1) = 5x + x²

Simplifying the equation, we get:

11 - 5x = 5x + x²

Rearranging the terms to form a quadratic equation, we have:

x² + 10x - 11 = 0

This is a quadratic equation in the form ax² + bx + c = 0, where a = 1, b = 10, and c = -11. To find the solutions for x, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Plugging in the values, we get:

x = (-10 ± √(10² - 4(1)(-11))) / (2(1))

x = (-10 ± √(100 + 44)) / 2

x = (-10 ± √144) / 2

x = (-10 ± 12) / 2

This gives us two possible values for x:

x₁ = (-10 + 12) / 2 = 1

x₂ = (-10 - 12) / 2 = -11

Now, we can substitute these values of x back into the equation y = -5x + 1 to find the corresponding values of y:

For x₁ = 1:

y₁ = -5(1) + 1 = -4

For x₂ = -11:

y₂ = -5(-11) + 1 = 56

Thus, we have two solutions to the system of equations:

(1, -4) and (-11, 56)

Another way to determine the number of solutions is by analyzing the discriminant of the quadratic equation. The discriminant (Δ) is the part of the quadratic formula under the square root: Δ = b² - 4ac. The discriminant tells us about the nature of the roots of the quadratic equation:

• If Δ > 0, the equation has two distinct real roots.
• If Δ = 0, the equation has one real root (a repeated root).
• If Δ < 0, the equation has no real roots.

In our case, the quadratic equation is x² + 10x - 11 = 0, so a = 1, b = 10, and c = -11. The discriminant is:

Δ = 10² - 4(1)(-11) = 100 + 44 = 144

Since Δ = 144 > 0, the quadratic equation has two distinct real roots, which confirms that the system of equations has two solutions.

In conclusion, by analyzing the system of equations 10 + y = 5x + x² and 5x + y = 1, we found that there are two possible solutions. This corresponds to the line intersecting the parabola at two distinct points. We arrived at this conclusion through both graphical interpretation and algebraic methods, including solving the quadratic equation and analyzing the discriminant. Understanding the relationship between the equations and their graphical representations provides valuable insights into the nature of solutions in systems of equations.

This exploration highlights the interplay between algebra and geometry in solving mathematical problems. The ability to visualize equations and understand their properties is crucial in determining the number of solutions and interpreting their meaning in a geometric context. In this specific case, the intersection of a parabola and a line showcases the rich possibilities and the importance of analytical techniques in solving systems of equations.