Solving Equations: Finding The Solution Set

Hey Plastik Magazine readers! Let's dive into some math and figure out how to solve a system of equations. Specifically, we're going to tackle the question: Which represents the solution(s) of the system of equations, $y=-x^2+6 x+16$ and $y=-4 x+37$? We'll find the solution set algebraically, which means we'll be using equations and calculations to pinpoint the exact points where these two equations meet. Don't worry, it's not as scary as it sounds! I'll walk you through it step-by-step, making it super clear and easy to follow. So grab your pencils and let's get started. Understanding how to solve these kinds of problems is essential for many areas in math and science, and it’s a great way to sharpen your problem-solving skills overall. Let's make this fun and engaging, just like everything else we do here at Plastik Magazine.

Understanding the Problem and Approach

Alright, guys, before we jump into the calculations, let's break down what we're actually trying to do. We've got two equations: $y = -x^2 + 6x + 16$ and $y = -4x + 37$. The first one, $y = -x^2 + 6x + 16$, is a quadratic equation, which means it will form a parabola when graphed. The second one, $y = -4x + 37$, is a linear equation, representing a straight line. What we're trying to find are the points where the parabola and the line intersect. These points are the solutions to the system of equations. Essentially, we are looking for the x and y values that satisfy both equations simultaneously. Graphically, this is where the two lines cross each other.

So, how do we find these points algebraically? We're going to use a method called substitution. Since both equations are already solved for $y$, we can set the expressions for $y$ equal to each other. This will give us a new equation with only one variable, $x$, which we can then solve. Once we find the values of $x$, we can plug them back into either of the original equations to find the corresponding $y$ values. These $(x, y)$ pairs will be our solution set. This method is incredibly versatile and works for many different types of equations, so it's a valuable tool to have in your mathematical toolkit. The key is to take it one step at a time and not to get overwhelmed by the equations. Think of it like a puzzle – each step brings you closer to the solution.

Step-by-Step Algebraic Solution

Now, let's get to the fun part: solving the equations! First, since we know that $y = -x^2 + 6x + 16$ and $y = -4x + 37$, we can set these two expressions equal to each other. This gives us: $-x^2 + 6x + 16 = -4x + 37$. This is our new equation, and our next step is to rearrange it into a standard quadratic form, which is $ax^2 + bx + c = 0$. To do this, we'll move all the terms to one side of the equation. Add $4x$ and subtract $37$ from both sides: $-x^2 + 6x + 4x + 16 - 37 = 0$. Combining like terms, we get $-x^2 + 10x - 21 = 0$. Now, it's usually easier to work with a positive $x^2$ term, so we can multiply the entire equation by $-1$, resulting in $x^2 - 10x + 21 = 0$.

Next, we need to solve this quadratic equation. We can do this by factoring. We are looking for two numbers that multiply to 21 and add up to -10. Those numbers are -3 and -7. Therefore, the factored form of the equation is $(x - 3)(x - 7) = 0$. For this equation to be true, either $(x - 3) = 0$ or $(x - 7) = 0$. Solving for $x$ in each case, we find that $x = 3$ or $x = 7$. We now have our $x$ values, and the next step is to find the corresponding $y$ values. We will substitute these $x$ values back into either of the original equations to solve for $y$. It is usually easier to use the linear equation, $y = -4x + 37$, but either will give you the correct answer. Now, let's move on to calculate the y values.

Finding the y-Values and the Solution Set

Alright, we've got our $x$ values: $x = 3$ and $x = 7$. Now, let's find the corresponding $y$ values. We'll start with $x = 3$. Plugging this into the equation $y = -4x + 37$, we get $y = -4(3) + 37$. Simplifying this gives us $y = -12 + 37$, so $y = 25$. This gives us our first solution point: $(3, 25)$. This means when $x$ is 3, $y$ is 25, and these values satisfy both equations. Next, we will do the same for $x = 7$. Plugging $x = 7$ into the equation $y = -4x + 37$, we have $y = -4(7) + 37$. Simplifying gives us $y = -28 + 37$, and therefore $y = 9$. This gives us our second solution point: $(7, 9)$. This means when $x$ is 7, $y$ is 9. So, the solution set for the system of equations consists of the points $(3, 25)$ and $(7, 9)$. These are the points where the parabola and the line intersect.

Therefore, the correct answer from the given options is C. $(3, 25)$ and $(7, 9)$. Remember, the solution set represents all points that satisfy both equations simultaneously. So, in this case, both $(3, 25)$ and $(7, 9)$ are the solutions to the system of equations. We've successfully determined the solution set algebraically, which means we have used equations and calculations to find the exact points of intersection. Great job, everyone! Let's do a quick recap to ensure that we got everything right.

Quick Recap and Final Answer

So, to recap what we did: we started with two equations, a quadratic and a linear equation. Then, we used substitution to set the two equations equal to each other. We rearranged this into a quadratic equation, and then we solved for x by factoring the quadratic. Once we had our x values, we substituted them back into the linear equation to solve for the corresponding y values. This gave us our two points of intersection: $(3, 25)$ and $(7, 9)$. Therefore, the solution set is $(3, 25)$ and $(7, 9)$, which corresponds to answer choice C. Great job, everyone! Remember, the key to solving systems of equations is to stay organized and take it one step at a time. This problem shows how important it is to be able to manipulate equations and apply different solution methods. This method can be applied to solve many problems in math. Keep practicing, and you'll get better and better at it. Keep an eye out for more math tutorials and articles here at Plastik Magazine. Thanks for reading, and see you next time!