Solving Systems Of Equations: Which Values To Find?

When tackling a system of equations, particularly one involving a quadratic and a linear equation like $y=3x^2-2x+18$ and $y=-x+15$, a key question arises: what values are we actually trying to find? This article will delve into the heart of solving systems of equations, highlighting the crucial role of identifying both the x and y values that satisfy all equations within the system. Understanding this fundamental concept is essential for mastering algebra and its applications in various fields.

Understanding Systems of Equations

At its core, a system of equations is a collection of two or more equations that share variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. Graphically, this solution represents the point(s) where the graphs of the equations intersect. In our specific case, we have a system consisting of a quadratic equation ($y=3x^2-2x+18$) and a linear equation ($y=-x+15$). The quadratic equation, when graphed, will form a parabola, while the linear equation will form a straight line. The points where these two graphs intersect represent the solutions to the system.

To truly grasp the solution, we can begin by considering what each equation represents on its own. The quadratic equation, $y = 3x^2 - 2x + 18$, traces a parabola in the coordinate plane. Each point on this parabola represents a pair of (x, y) values that satisfy the equation. Similarly, the linear equation, $y = -x + 15$, represents a straight line, with every point on the line corresponding to (x, y) values that make the equation true. Now, when we consider these equations as a system, we are looking for the specific points (x, y) that lie on both the parabola and the line. These points are the solutions to the system, because their x and y values satisfy both equations simultaneously. This understanding highlights the significance of finding both the x and y values, as they together define the points of intersection, which are the solutions we seek.

In the context of real-world problems, understanding systems of equations is invaluable. For example, in business, systems of equations can be used to model supply and demand curves, where the intersection point represents the market equilibrium price and quantity. In physics, these systems can describe the motion of objects under multiple forces. Therefore, the ability to solve these systems efficiently and accurately is not just an academic exercise but a crucial skill for various applications.

Why We Need Both x and y Values

When solving a system of equations, it's tempting to think that finding just one variable is sufficient. However, in most cases, particularly with systems involving different types of equations (like our quadratic and linear example), determining both the x and y values is crucial for a complete solution. The x value tells us where the intersection occurs horizontally on the graph, while the y value tells us where it occurs vertically. Only when we have both values do we know the exact coordinates of the solution point.

Consider our given equations: $y=3x^2-2x+18$ and $y=-x+15$. If we were to find only the x values, we would know the x-coordinates where the parabola and the line intersect. However, without the corresponding y values, we wouldn't know the exact points of intersection. Each x value could potentially correspond to a different y value on each curve. Therefore, simply knowing the x values isn't enough to pinpoint the solutions to the system.

To illustrate this further, suppose we only found an x value of, say, 1. Plugging x = 1 into the quadratic equation gives us $y = 3(1)^2 - 2(1) + 18 = 19$, and plugging it into the linear equation gives us $y = -1 + 15 = 14$. This clearly shows that at x = 1, the y values are different for the two equations, meaning (1,19) is a point on the parabola but not on the line, and (1,14) is a point on the line but not on the parabola. Therefore, x = 1 is not part of the solution to the system of equations. This example underscores the necessity of finding both x and y values to determine the points that satisfy both equations simultaneously.

Furthermore, in real-world applications, both x and y often represent distinct quantities. For instance, x might represent time, and y might represent distance. If we're modeling the motion of an object, knowing the time (x) at which two objects meet is just half the story; we also need to know the location (y) where they meet. Thus, finding both variables provides a complete and meaningful solution in practical contexts. In conclusion, solving a system of equations fully requires identifying both the x and y values to determine the exact points that satisfy all equations in the system.

Solving the System: A Step-by-Step Approach

To solve the system of equations $y=3x^2-2x+18$ and $y=-x+15$, we can use the method of substitution. Since both equations are expressed in terms of y, we can set them equal to each other:

$3x^2-2x+18 = -x+15$

Now, we rearrange the equation to form a quadratic equation in standard form:

$3x^2-2x+18 + x - 15 = 0$

$3x^2 - x + 3 = 0$

Next, we can use the quadratic formula to solve for x. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In our equation, $a = 3$, $b = -1$, and $c = 3$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(3)}}{2(3)}$

$x = \frac{1 \pm \sqrt{1 - 36}}{6}$

$x = \frac{1 \pm \sqrt{-35}}{6}$

Since the discriminant (the value inside the square root) is negative, there are no real solutions for x. This means that the parabola and the line do not intersect in the real coordinate plane. Therefore, there are no real solutions for this system of equations. However, if we had found real solutions for x, we would then substitute each x value back into either of the original equations to find the corresponding y values. This would give us the (x, y) pairs that represent the solutions to the system.

For example, let’s consider a modified system where the equations do intersect. Suppose we had found x values of x = 1 and x = 2. We could plug these values into the simpler linear equation, $y = -x + 15$, to find the corresponding y values. For x = 1, we get $y = -1 + 15 = 14$, and for x = 2, we get $y = -2 + 15 = 13$. Therefore, the solutions to this hypothetical system would be the points (1, 14) and (2, 13). This step-by-step approach illustrates the importance of finding both x and y values to fully solve the system of equations, emphasizing that each x value must be paired with its corresponding y value to represent a point of intersection.

Conclusion

In conclusion, when solving a system of equations, particularly one involving a quadratic and a linear equation, it is essential to find both the x and y values. These values represent the coordinates of the points where the graphs of the equations intersect, which are the solutions to the system. Understanding this concept is fundamental to solving algebraic problems and applying these skills in various real-world scenarios.

Therefore, the correct answer to the question is C. Both x and y values. Finding both values provides a complete and accurate solution to the system of equations, giving us the precise coordinates of the intersection points.

For further exploration of systems of equations and related topics, consider visiting Khan Academy's Algebra section. You'll find a wealth of resources, including videos, practice exercises, and articles, to help you deepen your understanding.