Solve System Equations: Step-by-Step Guide
Hey guys! Have you ever stared at a pair of equations and felt like you're trying to crack a secret code? Well, you're not alone! Solving systems of equations can seem daunting, but it's actually a super useful skill in math and beyond. In this article, we're diving deep into one such system, exploring the ins and outs of how to find the solutions. We'll break down the problem step-by-step, making sure you understand not just the answer, but why it's the answer. Our focus today is on the system:
But before we jump into the nitty-gritty, let's chat a bit about what a "solution" actually means in this context. Think of it like this: we're looking for the points (x, y) that satisfy both equations simultaneously. These points are where the graphs of the two equations intersect. So, we're not just finding numbers; we're finding coordinates on a graph! In this article, we will guide you through the solution of the given system of equations, providing a clear and understandable explanation. We will cover the algebraic steps involved in finding the intersection points, ensuring a solid grasp of the methods used. By understanding these principles, you'll be well-equipped to tackle similar problems with confidence. So, grab your pencils, and let's get started on this mathematical adventure!
Okay, let's break down these equations. The first one, $y = -(x+2)^2 + 1$, might look a little intimidating, but it's actually a parabola. Remember those U-shaped curves? The negative sign in front of the parentheses tells us it opens downwards, and the "+2" inside the parentheses shifts it to the left by 2 units. The "+1" at the end moves it up 1 unit. So, we've got a downward-facing parabola with its peak (vertex) at the point (-2, 1). The second equation, $y = 2x + 5$, is much simpler – it's a straight line. The "2" is the slope (how steep the line is), and the "+5" is the y-intercept (where the line crosses the y-axis). To truly understand the solutions, visualize these two graphs intersecting. The points of intersection are the solutions we seek. Imagine the parabola, a gentle curve reaching its peak and descending, and the straight line, steadily climbing upwards. They might meet at one point, two points, or even not at all! The solutions to the system will be the coordinates of any intersection points. This graphical perspective is crucial as it provides a visual confirmation of our algebraic solutions. Understanding the shapes and positions of these graphs not only helps in solving the equations but also in predicting the nature and number of solutions. So, as we dive into the algebraic manipulations, keep this visual picture in mind. It's a powerful tool for making sense of the math!
Alright, enough with the visuals – let's get our hands dirty with some algebra! The key to solving this system is to realize that both equations are equal to y. That means we can set them equal to each other. So, we have:
Now, we've got one equation with just one variable (x), which is something we can work with! First, let's expand that squared term:
Distribute the negative sign:
Simplify things a bit:
Now, let's get everything on one side to form a quadratic equation. Add $x^2$, $4x$, and 3 to both sides:
We now have a quadratic equation in the standard form, which we can solve by factoring, completing the square, or using the quadratic formula. Factoring is often the quickest method if possible. In this case, we're looking for two numbers that multiply to 8 and add to 6. Those numbers are 4 and 2. So, we can factor the equation as:
Setting each factor equal to zero gives us the solutions for x:
Now that we have the x-values, we can plug them back into either of the original equations to find the corresponding y-values. The linear equation $y = 2x + 5$ is simpler, so let's use that.
For $x = -4$:
For $x = -2$:
So, we've found our solutions! The points of intersection are (-4, -3) and (-2, 1). This algebraic journey, with its careful steps and transformations, has led us to the precise coordinates where the parabola and line meet. It’s a testament to the power of algebra in unveiling the hidden solutions within equations.
Before we celebrate our victory, it's always a good idea to double-check our work. We've found two potential solutions: (-4, -3) and (-2, 1). To make sure these are correct, we'll plug them back into both original equations. If they satisfy both, we're golden!
Let's start with (-4, -3). For the parabola equation:
That checks out! Now for the line equation:
Perfect! (-4, -3) works in both equations. Let's do the same for (-2, 1). For the parabola equation:
Great! And for the line equation:
Double perfect! Both points satisfy both equations. Verifying our solutions is a crucial step in the problem-solving process. It ensures that our algebraic manipulations have been accurate and that we haven't made any errors along the way. This meticulous approach not only gives us confidence in our answers but also reinforces the importance of precision in mathematics.
As we discussed earlier, visualizing the equations can provide a powerful confirmation of our algebraic solutions. Imagine the parabola $y = -(x+2)^2 + 1$, opening downwards with its vertex at (-2, 1), and the line $y = 2x + 5$, steadily rising. If we were to graph these two equations, we would see them intersect at the points (-4, -3) and (-2, 1). This graphical representation serves as a visual confirmation of our algebraic solutions. By plotting these equations, we can see the points of intersection clearly, reinforcing our understanding that these are indeed the solutions to the system. This graphical verification is not just a check; it's a way to connect the algebraic manipulations with the geometric representation, enhancing our overall comprehension of the problem.
After all our hard work, we've arrived at the solutions: (-4, -3) and (-2, 1). Now, let's look at the answer choices provided:
A. (-4, -3) and (-2, 1) B. (-4, -3) and (2, 1) C. (-4, 5) and (2, 1) D. Discussion category
It's clear that option A, (-4, -3) and (-2, 1), is the correct answer. We've successfully navigated the algebraic steps, verified our solutions, and even considered the graphical representation. This thorough approach ensures that we not only arrive at the correct answer but also understand the underlying concepts. By breaking down the problem into manageable steps and using a combination of algebraic and graphical methods, we've demonstrated a comprehensive understanding of solving systems of equations.
So, there you have it! We've successfully solved the system of equations and found the solutions to be (-4, -3) and (-2, 1). But more importantly, we've explored the process of solving these kinds of problems. We talked about understanding the equations, using algebra to find the solutions, verifying our answers, and even visualizing the graphs. Solving systems of equations is a fundamental skill in mathematics, with applications in various fields, from engineering to economics. By mastering this skill, you're not just solving equations; you're developing a way of thinking that is crucial for problem-solving in many areas of life. Remember, the key is to break down the problem into manageable steps, use the tools you have (algebra, graphs, verification), and don't be afraid to ask for help when you need it. With practice and persistence, you'll be solving systems of equations like a pro in no time! Keep practicing, keep exploring, and most importantly, keep enjoying the journey of mathematical discovery.
