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

by Andrew McMorgan 52 views

Hey guys! Ever find yourselves staring blankly at a system of equations, wondering where to even begin? Don't sweat it! We're going to break down a super common problem and walk through it together. Today, we're tackling the system:

y = 220x - 160
y = 20x^2 - 400

Ready? Let's dive in!

Understanding the Problem

Before we jump into solving, let's quickly understand what we're looking at. A system of equations is just a set of two or more equations that we want to solve simultaneously. In other words, we're trying to find the values of x and y that make both equations true at the same time. Think of it like finding the point where two lines or curves intersect on a graph. This intersection point represents the solution to our system. The first equation, y = 220x - 160, represents a straight line. The second equation, y = 20x^2 - 400, represents a parabola. Our goal is to find the x and y coordinates where this line and parabola meet. This method will help in various fields such as engineering, economics, and computer science.

Now, why is this important? Well, systems of equations pop up everywhere in real-world problems. Imagine you're trying to figure out the break-even point for a business, or modeling the trajectory of a rocket, or even optimizing a recipe! Knowing how to solve these systems is a seriously valuable skill. We solve systems of equations to find equilibrium points in supply and demand curves in economics. In physics, we can use them to analyze forces acting on an object. In computer graphics, they help determine intersections of lines and surfaces for rendering images. So, understanding these concepts not only helps in math class but also in many practical applications.

Step 1: Setting the Equations Equal

The key to solving this particular system is recognizing that both equations are already solved for y. This means we can set the right-hand sides of the equations equal to each other. This is because, at the point where the line and parabola intersect, the y values must be the same. Therefore, we can write:

220x - 160 = 20x^2 - 400

What we've done here is transform the problem from a system of two equations into a single equation with just one variable, x. This is a huge step forward! Setting the equations equal allows us to eliminate one variable and solve for the other. This technique is commonly used when both equations are expressed in terms of the same variable. It simplifies the problem and makes it easier to find the solution. By equating the expressions, we are essentially looking for the x-values where the two functions have the same y-value, representing the points of intersection. This algebraic manipulation is a powerful tool in solving systems of equations.

Step 2: Rearranging into a Quadratic Equation

Now we have a single equation: 220x - 160 = 20x^2 - 400. To solve for x, we need to rearrange this into a standard quadratic equation form, which is ax² + bx + c = 0. Let's do that:

  1. Subtract 220x from both sides: -160 = 20x^2 - 220x - 400
  2. Add 160 to both sides: 0 = 20x^2 - 220x - 240

Now our equation looks like this: 20x^2 - 220x - 240 = 0. A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable we want to solve for. Quadratic equations appear frequently in mathematics and physics, describing various phenomena such as projectile motion, areas, and optimization problems. The most common methods for solving quadratic equations are factoring, completing the square, and using the quadratic formula. The solutions to a quadratic equation are also known as its roots or zeros, representing the values of x that satisfy the equation.

Step 3: Simplifying the Quadratic Equation

Notice that all the coefficients in our equation 20x^2 - 220x - 240 = 0 are divisible by 20. Dividing the entire equation by 20 will simplify it and make it easier to work with:

x^2 - 11x - 12 = 0

This simplified equation is much easier to factor or solve using the quadratic formula. Simplifying equations before solving is always a good practice, as it reduces the complexity and the chance of making errors. Dividing by a common factor doesn't change the solutions of the equation, but it makes the numbers smaller and more manageable. In this case, dividing by 20 transformed the equation into a more straightforward form, allowing us to proceed with factoring more easily. This step is crucial for efficient problem-solving in algebra and beyond.

Step 4: Factoring the Quadratic Equation

Now we have the simplified quadratic equation: x^2 - 11x - 12 = 0. Let's try to factor it. We're looking for two numbers that multiply to -12 and add up to -11. Those numbers are -12 and +1.

So, we can factor the equation as follows:

(x - 12)(x + 1) = 0

Factoring involves breaking down a mathematical expression into a product of simpler expressions. In the context of quadratic equations, factoring helps us find the roots or solutions of the equation. The technique relies on identifying two numbers that satisfy specific conditions related to the coefficients of the quadratic equation. For example, in the equation x^2 - 11x - 12 = 0, we seek two numbers that multiply to -12 and add up to -11. Once we find these numbers, we can rewrite the quadratic equation as a product of two binomials. Factoring is a powerful and efficient method for solving quadratic equations, especially when the roots are integers or simple fractions. However, not all quadratic equations can be easily factored, in which case other methods like the quadratic formula must be used.

Step 5: Solving for x

To find the solutions for x, we set each factor equal to zero:

x - 12 = 0  or  x + 1 = 0

Solving these simple equations gives us:

x = 12  or  x = -1

So, we have two possible values for x: 12 and -1. These are the x-coordinates of the points where the line and parabola intersect. Each value of x represents a solution to the equation, indicating where the function crosses the x-axis or, in the context of a system of equations, where the curves intersect. Finding these values is a fundamental step in solving equations and understanding the behavior of functions. Whether it's a simple linear equation or a complex polynomial equation, the goal remains the same: to isolate the variable and determine the values that satisfy the equation. These solutions provide critical insights into the problem being modeled and are essential for making informed decisions based on mathematical analysis.

Step 6: Solving for y

Now that we have the values of x, we need to find the corresponding values of y. We can use either of the original equations to do this. Let's use the first equation, y = 220x - 160, because it's simpler:

For x = 12:

y = 220(12) - 160
y = 2640 - 160
y = 2480

For x = -1:

y = 220(-1) - 160
y = -220 - 160
y = -380

So, we have two solutions for y: 2480 and -380.

Solving for y involves substituting the known value of x back into one of the original equations to find the corresponding value of y. This step is crucial because it completes the solution, providing both the x and y coordinates that satisfy the equation or system of equations. By plugging in the x value, we can isolate y and perform the necessary calculations to determine its value. This process is fundamental in various mathematical and scientific applications, allowing us to find specific points on a curve, determine relationships between variables, and solve real-world problems. In the context of systems of equations, solving for y helps us find the points of intersection between different functions, which are essential for understanding the behavior of complex systems.

Step 7: State the Solutions

Finally, we can state the solutions to the system of equations. We have two solutions, which are the points where the line and parabola intersect:

  • (12, 2480)
  • (-1, -380)

These are the pairs of x and y values that satisfy both equations in the system. Stating the solutions clearly is crucial for conveying the results of a mathematical problem accurately. It involves summarizing the values of the variables that satisfy the given equation or system of equations. For example, in a system of equations, the solutions are the ordered pairs (x, y) that make all the equations true simultaneously. Presenting the solutions in a clear and organized manner allows others to easily understand the results and their implications. This step is not just about providing the numerical answers but also about ensuring that the solutions are correctly interpreted and applied in the appropriate context. Whether in algebra, calculus, or differential equations, clearly stating the solutions is a fundamental aspect of mathematical communication.

Conclusion

And there you have it! We've successfully solved the system of equations. Remember, the key steps were:

  1. Setting the equations equal to each other.
  2. Rearranging into a quadratic equation.
  3. Simplifying the quadratic equation.
  4. Factoring the quadratic equation.
  5. Solving for x.
  6. Solving for y.
  7. Stating the solutions.

Keep practicing, and you'll become a system-of-equations-solving pro in no time! You got this!

Solving systems of equations is a fundamental skill in mathematics with wide-ranging applications across various fields. From determining equilibrium points in economics to analyzing forces in physics and optimizing computer graphics, the ability to find solutions to these systems is invaluable. Whether it involves linear equations, quadratic equations, or more complex functions, the underlying principles remain the same: manipulate the equations to isolate variables and find values that satisfy all conditions simultaneously. Mastering these techniques not only enhances problem-solving abilities but also provides a deeper understanding of the relationships between different variables and the models they represent. So, keep honing your skills, explore different methods, and embrace the power of systems of equations in unlocking solutions to real-world challenges.