Solving Systems Of Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of solving systems of equations. This is a super important concept in algebra, and understanding it will give you a solid foundation for more advanced math. Today, we're going to break down how to figure out how many solutions a system of equations can have. So, buckle up; this is going to be fun and informative!

Understanding Systems of Equations: The Basics

First off, what exactly is a system of equations? Well, it's simply a set of two or more equations that we want to solve simultaneously. Think of it like a puzzle where we need to find the values of the variables (usually x and y) that satisfy all the equations in the system. These values represent the point(s) where the graphs of the equations intersect. Each equation in the system represents a line or curve, and the solutions are the points where these lines or curves meet. A system can have no solution, one solution, or infinitely many solutions. This all depends on the nature of the equations and how they relate to each other graphically.

Let’s use a simple analogy: imagine you have two roads on a map. Each road represents an equation. The solution to the system is where those roads intersect. If the roads are parallel, they never intersect, meaning there's no solution. If the roads are the same (overlapping), they intersect everywhere, so there are infinite solutions. If the roads cross at one point, there's only one solution. Pretty cool, right? Now, let's talk about the specific system of equations we're dealing with:

$\begin{array}{l} 10+y=5 x+x^2 \ 5 x+y=1 \ \end{array}$

This system involves a quadratic equation and a linear equation. The quadratic equation, $10 + y = 5x + x^2$, represents a parabola, while the linear equation, $5x + y = 1$, represents a straight line. Finding the number of solutions means determining how many times the line intersects the parabola. The intersection points give us the values of x and y that satisfy both equations simultaneously. So, our main goal is to figure out whether the line and parabola cross each other at all, at one point, or at multiple points. Let's get cracking and solve this problem step by step!

Breaking Down the Equations

Let's take a closer look at our system of equations. We have:

1. Equation 1: $10 + y = 5x + x^2$ (This is a quadratic equation; it will create a parabola when graphed).
2. Equation 2: $5x + y = 1$ (This is a linear equation; it will create a straight line when graphed).

To figure out the number of solutions, we can use different methods. One way is to solve the system algebraically, which will give us the exact values of x and y (if there are any solutions). Another approach is to think about the graphs of the equations. The number of solutions corresponds to the number of intersection points between the parabola and the line. Remember, intersections are the x and y values that make both equations true at the same time. The way the equations are set up gives us a clue about the potential solutions. The quadratic equation hints that we might find zero, one, or two solutions. This is because a line can intersect a parabola in these ways. Now, let's proceed to determine the exact number of solutions using the most efficient methods.

Solving the System Algebraically

Alright, let’s get down to the nitty-gritty and solve this system algebraically. This approach gives us the most precise way to find out how many solutions there are. Here’s how we can do it:

1. Isolate y in the linear equation: From Equation 2 ($5x + y = 1$), we can easily isolate y: $y = 1 - 5x$.
2. Substitute into the quadratic equation: Now, substitute the expression for y (which is $1 - 5x$) into Equation 1: $10 + (1 - 5x) = 5x + x^2$
3. Simplify and rearrange to get a quadratic equation: Combine like terms: $11 - 5x = 5x + x^2$ Move everything to one side to get a standard quadratic form: $0 = x^2 + 10x - 11$
4. Solve the quadratic equation: We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's try factoring, since it's often the easiest method when it works. We're looking for two numbers that multiply to -11 and add up to 10. Those numbers are 11 and -1. So, we can factor the equation as: $(x + 11)(x - 1) = 0$ This means either $(x + 11) = 0$ or $(x - 1) = 0$. Solving for x gives us: $x = -11$ or $x = 1$
5. Find the corresponding y values: Now that we have the x values, we can plug them back into the linear equation ($y = 1 - 5x$) to find the corresponding y values.
   • For $x = -11$: $y = 1 - 5(-11) = 1 + 55 = 56$
   • For $x = 1$: $y = 1 - 5(1) = 1 - 5 = -4$ So, the solutions are $(-11, 56)$ and $(1, -4)$.

Analyzing the Solutions and Answering the Question

Great job, guys! We have successfully solved the system of equations. We found two distinct solutions, which are the points $(-11, 56)$ and $(1, -4)$. These points represent the intersection of the parabola and the straight line. Since we have two distinct solutions, we can confidently say that the system of equations has two possible solutions. The answer choice that matches this is C. 2. This process not only solves the problem but also demonstrates the importance of algebraic methods in finding exact solutions. The solutions we derived precisely locate the points where the parabola and the line intersect, showcasing the power of solving systems of equations. Understanding the step-by-step process is crucial for tackling similar problems in the future.

Visualizing the Solutions: A Graphical Perspective

To make things even clearer, let's visualize this system graphically. If we were to plot both the parabola and the line on a coordinate plane, the points where they intersect would be our solutions. Remember that the solutions to a system of equations are the points where the graphs of the equations intersect. Graphing helps us visually confirm our algebraic solutions. The parabola ($y = x^2 + 5x - 10$) and the line ($y = 1 - 5x$) will intersect at two points. Plotting these equations can provide an intuitive understanding of the solution, helping us grasp the geometric meaning of the algebraic results. This visual approach can be beneficial in confirming the number of solutions and understanding the relationships between the equations.

1. Graphing the Equations:
   • The Parabola: The equation $10 + y = 5x + x^2$ can be rewritten as $y = x^2 + 5x - 10$. This is a parabola that opens upwards. You can plot this by finding the vertex and a few other points. The vertex of the parabola is at $x = -b/2a = -5/2$. The corresponding y-value is $y = (-5/2)^2 + 5(-5/2) - 10 = -6.25 - 12.5 - 10 = -18.75$. So, the vertex is at $(-2.5, -18.75)$.
   • The Line: The equation $5x + y = 1$ can be rewritten as $y = 1 - 5x$. This is a straight line with a slope of -5 and a y-intercept of 1. You can plot this by finding two points, such as $(0, 1)$ and $(1, -4)$.
2. Identifying the Intersections: When you graph these two equations, you’ll see that the line intersects the parabola at two points: $(-11, 56)$ and $(1, -4)$, which matches our algebraic solutions!

This confirms that our algebraic solution is correct and that the system has two solutions. This graphical approach is a powerful tool to verify your answers and gain a deeper understanding of the problem.

Quick Recap and Key Takeaways

Alright, let’s quickly recap what we’ve covered today, because it's super important to remember the key points.

• Systems of Equations: A set of two or more equations that we solve simultaneously.
• Solutions: The values of the variables that satisfy all equations in the system.
• Number of Solutions: A system can have zero, one, or infinitely many solutions.
• Our Example: We solved the system $\begin{array}{l} 10+y=5 x+x^2 \ 5 x+y=1 \ \end{array}$ and found two solutions, which means the line and parabola intersect at two points.
• Methods Used: We used both algebraic (substitution and solving the resulting quadratic equation) and graphical methods to find and visualize the solutions.

Why This Matters

Understanding how to solve systems of equations is fundamental in algebra and beyond. This skill is used in numerous applications, from basic math problems to complex scientific and engineering models. Being able to solve these types of problems provides you with problem-solving skills and critical thinking, which are essential in various fields.

• Real-World Applications: Solving systems of equations is used in many real-world scenarios, such as in economics to determine market equilibrium, in physics to analyze the motion of objects, and in computer science to develop algorithms.
• Future Math Courses: This knowledge is a prerequisite for more advanced topics in math, such as calculus and linear algebra.

So, by mastering this skill, you're not just answering a math question; you're building a foundation for future success. Keep practicing, keep learning, and you'll do great! And that concludes our journey through solving this system of equations! Remember, practice makes perfect. Keep up the great work, and happy solving!