Determining Real Solutions For Systems Of Equations A Comprehensive Analysis

In the realm of mathematics, systems of equations play a crucial role in modeling real-world phenomena and solving complex problems. A system of equations is a collection of two or more equations with the same set of variables. The solutions to a system of equations are the values of the variables that satisfy all equations simultaneously. Determining the number of real solutions for a given system is a fundamental task that provides insights into the nature of the relationships between the equations. In this article, we will delve into the intricacies of analyzing systems of equations and explore various techniques for determining the number of real solutions.

Let's begin by examining System A, which consists of two equations:

• $x^2 + y^2 = 17$
• y = -rac{1}{2}x

The first equation represents a circle centered at the origin (0, 0) with a radius of $\sqrt{17}$. The second equation represents a line with a slope of -1/2 passing through the origin. To determine the number of real solutions, we need to find the points where the circle and the line intersect.

Substituting the second equation into the first equation, we get:

$x^2 + (-\frac{1}{2}x)^2 = 17$

Simplifying the equation, we have:

$x^2 + \frac{1}{4}x^2 = 17$

$\frac{5}{4}x^2 = 17$

$x^2 = \frac{68}{5}$

$x = \pm\sqrt{\frac{68}{5}}$

This gives us two distinct real values for x. For each value of x, we can substitute it back into the equation y = -rac{1}{2}x to find the corresponding y-value. Therefore, System A has two real solutions. These solutions represent the points where the line intersects the circle.

To gain a deeper understanding, we can visualize the system graphically. The circle and the line intersect at two points, confirming our algebraic solution. The coordinates of these points represent the real solutions to the system of equations. In this case, the algebraic manipulation, where we substitute one equation into another, allows us to reduce the system to a single equation in one variable. The solutions of this reduced equation then lead us to the solutions of the original system. This technique is widely applicable in solving various systems of equations, especially those involving a mix of linear and non-linear equations.

Furthermore, the discriminant of the resulting quadratic equation can be used to determine the number of real solutions without explicitly solving for them. However, in this case, solving for x is straightforward and provides a clear picture of the intersection points.

Now, let's consider System B, which comprises the following equations:

• $y = x^2 - 7x + 10$
• $y = -6x + 5$

The first equation represents a parabola, while the second equation represents a line. To find the number of real solutions, we need to determine the points where the parabola and the line intersect.

Setting the two equations equal to each other, we get:

$x^2 - 7x + 10 = -6x + 5$

Rearranging the equation, we have:

$x^2 - x + 5 = 0$

This is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 1$, $b = -1$, and $c = 5$. To determine the number of real solutions, we can use the discriminant, which is given by:

$\Delta = b^2 - 4ac$

Substituting the values, we get:

$\Delta = (-1)^2 - 4(1)(5) = 1 - 20 = -19$

Since the discriminant is negative, the quadratic equation has no real roots. This means that the parabola and the line do not intersect in the real plane. Therefore, System B has no real solutions.

The negative discriminant indicates that the roots of the quadratic equation are complex numbers, implying that there are no real values of x that satisfy the equation. Graphically, this means that the line and the parabola do not intersect. The line either lies entirely above or entirely below the parabola without any common points. This type of analysis using the discriminant is a powerful tool for quickly determining the nature of the solutions to a quadratic equation and, consequently, for understanding the intersection behavior of the curves represented by the system of equations.

The concept of the discriminant is not limited to quadratic equations alone; similar discriminants exist for higher-degree polynomials, although their calculation and interpretation become more complex. In general, the sign of the discriminant provides valuable information about the number and nature of the roots of a polynomial equation.

Finally, let's analyze System C, which consists of the equations:

• $y = -2x^2 + 9$
• $8x - y = -17$

The first equation represents a parabola opening downwards, and the second equation represents a line. To find the number of real solutions, we need to find the points where the parabola and the line intersect.

Solving the second equation for y, we get:

$y = 8x + 17$

Substituting this into the first equation, we have:

$8x + 17 = -2x^2 + 9$

Rearranging the equation, we get:

$2x^2 + 8x + 8 = 0$

Dividing the equation by 2, we simplify it to:

$x^2 + 4x + 4 = 0$

This is a quadratic equation that can be factored as:

$(x + 2)^2 = 0$

This gives us a single real solution for x:

$x = -2$

Substituting this value back into the equation $y = 8x + 17$, we get:

$y = 8(-2) + 17 = 1$

Therefore, System C has one real solution, which is the point (-2, 1). This indicates that the line is tangent to the parabola at this point. The tangency implies that the line touches the parabola at exactly one point, which corresponds to the single real solution we found.

In this case, the fact that the quadratic equation has a repeated root (or a double root) is directly related to the geometric interpretation of the line being tangent to the parabola. A repeated root signifies that the discriminant of the quadratic equation is zero. This is a key concept in understanding the relationship between algebraic solutions and geometric intersections.

When solving systems of equations, if a quadratic equation arises and its discriminant is zero, it is a strong indicator that the curves represented by the equations are tangent to each other. This connection between the discriminant and the geometry of the curves provides valuable insights into the nature of the solutions.

In this article, we have explored the process of determining the number of real solutions for systems of equations. We analyzed three different systems, each consisting of two equations. By using algebraic techniques such as substitution and the discriminant, we were able to determine the number of real solutions for each system. System A had two real solutions, System B had no real solutions, and System C had one real solution. The number of real solutions corresponds to the number of intersection points between the graphs of the equations in the system. Understanding how to analyze systems of equations is a fundamental skill in mathematics and has wide-ranging applications in various fields, including physics, engineering, and economics.

The techniques demonstrated here, such as substitution and discriminant analysis, are not only useful for determining the number of solutions but also for finding the solutions themselves. Furthermore, the graphical interpretation of the solutions provides a visual understanding of the relationships between the equations in the system. The ability to connect algebraic solutions with geometric representations is a crucial aspect of mathematical thinking and problem-solving.

In summary, the analysis of systems of equations is a multifaceted process that combines algebraic manipulation, geometric visualization, and a solid understanding of fundamental mathematical concepts. The number of real solutions to a system provides important information about the nature of the relationships between the equations, and the techniques for determining this number are essential tools in the mathematical arsenal.