Solving Systems Of Equations Identifying Equations From A System

Let's dive into how we can figure out which equation can be solved using the given system of equations. This involves understanding how systems of equations work and how they relate to individual equations. We'll break it down step by step to make it super clear.

Understanding the System of Equations

First, let's take a look at the system of equations we're working with:

\left\{\begin{array}{l}
y=3 x^5-5 x^3+2 x^2-10 x+4 \\
y=4 x^4+6 x^3-11
\end{array}\right.

This system consists of two equations, both of which express y in terms of x. The first equation is a quintic polynomial (a polynomial of degree 5), and the second equation is a quartic polynomial (a polynomial of degree 4). When we have a system like this, we're essentially looking for the values of x and y that satisfy both equations simultaneously. Think of it as finding the points where the graphs of these two equations intersect. These intersection points represent the solutions to the system.

How to Solve Systems of Equations

There are a few ways we can solve systems of equations, but in this case, the most straightforward approach is to use the substitution method. Since both equations are already solved for y, we can set them equal to each other. This is because, at the points where the two equations intersect, the y-values must be the same. So, we're essentially saying that if y equals one expression in terms of x, and y also equals another expression in terms of x, then those two expressions must be equal to each other at the solution points. This is a fundamental concept in solving systems of equations, and it's super useful in various mathematical contexts.

By setting the two expressions for y equal, we eliminate y and end up with a single equation in terms of x. This new equation represents the condition that must be true for the x-values at the intersection points. Solving this equation will give us the x-coordinates of the solutions. Once we have the x-values, we can plug them back into either of the original equations to find the corresponding y-values. This will give us the complete solution set for the system.

Now, let's actually perform the substitution and see what equation we get. This will lead us directly to the equation that can be solved using this system. It's like connecting the dots to reveal the bigger picture, and it's a crucial step in understanding how these equations relate to each other.

Setting the Equations Equal

To find the equation that can be solved using this system, we set the two expressions for y equal to each other:

3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11

This equation is formed by equating the right-hand sides of the two original equations. Remember, we're doing this because we're looking for the points where the y-values are the same for both equations. This step is crucial because it transforms the system of two equations into a single equation that we can solve for x. This is a common technique in algebra and is used extensively in various problem-solving scenarios. By setting the equations equal, we're essentially finding the x-values where the two curves represented by the original equations intersect. This intersection represents the solution to the system, and the resulting equation captures the condition that must be met at these points.

Simplifying the Equation

Now, let's simplify this equation to make it look more manageable. We want to get all the terms on one side, so we'll subtract 4x^4 + 6x^3 - 11 from both sides:

3x^5 - 5x^3 + 2x^2 - 10x + 4 - (4x^4 + 6x^3 - 11) = 0

This step is about rearranging the terms to create a standard form equation that's easier to work with. Think of it as organizing your workspace before tackling a big project. By moving all the terms to one side, we set the stage for further simplification and eventual solution. This is a common practice in algebra and calculus, where manipulating equations into standard forms is often the first step in solving them. The goal here is to consolidate like terms and create a single polynomial expression that equals zero. This form is particularly useful because it allows us to use techniques like factoring or numerical methods to find the roots, which are the solutions to the equation.

Next, we combine like terms:

3x^5 - 4x^4 - 11x^3 + 2x^2 - 10x + 15 = 0

This simplified equation represents the condition that must be satisfied for the x-values at the intersection points of the two original equations. It's a quintic polynomial equation, which means it's a polynomial equation of degree 5. Solving this equation can be challenging, but it's the key to finding the x-values that satisfy both equations in the original system. This equation encapsulates the relationship between the two original equations and allows us to focus on finding the values of x that make this entire expression equal to zero. These values of x are the roots of the polynomial, and they correspond to the x-coordinates of the points where the graphs of the two original equations intersect.

Identifying the Correct Equation

Now, let's compare our simplified equation with the given options. The options are:

• A. 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 0
• B. 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11

Our simplified equation is:

3x^5 - 4x^4 - 11x^3 + 2x^2 - 10x + 15 = 0

Notice that option B, 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11, is the equation we obtained before simplifying. This is the equation we got by setting the two original equations equal to each other. So, this is the equation that can be solved using the system of equations.

Why Option B is the Key

Option B represents the direct result of equating the two expressions for y in the original system. This step is crucial because it transforms the system of two equations into a single equation that we can solve for x. Think of it as finding the common ground between the two equations. The solutions to this equation are the x-coordinates of the points where the graphs of the two original equations intersect. These intersection points are the solutions to the system, so finding them is our primary goal. Option B encapsulates this fundamental relationship and provides a pathway to solving the system.

Option A, on the other hand, only represents the first equation in the system set equal to zero. While finding the roots of this equation might give us some interesting information about the first equation, it doesn't directly address the system as a whole. It's like focusing on a single piece of a puzzle rather than the entire picture. Option A doesn't take into account the second equation, which is essential for finding the solutions that satisfy both equations simultaneously. Therefore, while option A is a valid equation, it's not the equation that we can solve using the system of equations.

Final Answer

Therefore, the correct answer is B. The equation 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11 can be solved by using the given system of equations.

Which equation can be solved using the given system of equations?

Solving Systems of Equations Identifying Equations from a System ️