Solving Equations: A Step-by-Step Guide

Nov 6, 2025 by Admin 40 views

Hey math enthusiasts! Today, we're diving into a classic problem: solving a system of equations. This is super important stuff, whether you're a student just starting out or someone brushing up on their algebra skills. We'll break down how to tackle the specific problem given and explore the concepts involved, so you can totally nail it. Let's get started!

Understanding the Problem: What Does Solving a System Mean?

Okay, so what exactly does it mean to "solve a system of equations"? Well, in this context, we have two equations, and we want to find the point (or points) where they intersect. Think of each equation as a line on a graph. The solution to the system is the point (or points) where those lines cross each other. This point satisfies both equations simultaneously. It's like finding a treasure where both maps agree! The given system is:

$y = x - 2$

$y = x^2 - 3x + 2$

The first equation, $y = x - 2$ , represents a straight line. The second equation, $y = x^2 - 3x + 2$ , represents a parabola (a U-shaped curve). Our goal is to find the x and y values that make both of these equations true at the same time. The question provides us with multiple choices, and we will be selecting the answer that satisfies the conditions. So, let's look at how to approach this problem and find the correct solution from the given options.

To solve this, we can use a method called substitution. Since we know that $y$ is equal to $x - 2$ , we can substitute $(x-2)$ for $y$ in the second equation. This will give us a new equation with only one variable ( $x$ ), which we can then solve. This approach simplifies the problem and allows us to find the specific values of $x$ and $y$ that simultaneously satisfy both the equations. Let's dive deeper into how to perform this substitution and get to our solution!

The Substitution Method: Your Secret Weapon

Alright, let's get down to the nitty-gritty of solving this system using the substitution method. As we mentioned, we're going to replace $y$ in the second equation with what it's equal to in the first equation, which is $x - 2$ . This will give us a single equation we can solve for $x$ .

So, the second equation is $y = x^2 - 3x + 2$ . We substitute $x - 2$ for $y$ :

$x - 2 = x^2 - 3x + 2$

Now, we've got a quadratic equation. To solve it, we need to get everything on one side of the equation and set it equal to zero. Let's move all the terms to the right side:

$0 = x^2 - 3x + 2 - x + 2$

Simplifying, we get:

$0 = x^2 - 4x + 4$

Now, we can solve this quadratic equation. You can do this by factoring, using the quadratic formula, or completing the square. In this case, factoring works nicely. We're looking for two numbers that multiply to 4 and add up to -4. Those numbers are -2 and -2.

So, the factored equation is:

$0 = (x - 2)(x - 2)$ or $0 = (x-2)^2$

This means that $x - 2 = 0$ , so $x = 2$ . We've found our $x$ value! But remember, the solution to a system of equations is a point, which means we need both an $x$ and a $y$ value. This is where it's important to keep track of every step. We have effectively simplified the given question and are now at the crucial point of the solution.

Now that we've found our x-value, we can solve for y and determine which of the multiple-choice options is correct. Let's move to the next step, where we'll plug in the value of x into one of our initial equations and determine the y coordinate.

Finding the y-Value: The Final Piece of the Puzzle

Okay, we've found that $x = 2$ . Now, let's find the corresponding $y$ value. We can plug this $x$ value into either of the original equations. It's usually easiest to use the simpler one, which is $y = x - 2$ .

So, substitute $x = 2$ into $y = x - 2$ :

$y = 2 - 2$

$y = 0$

Therefore, the solution to the system of equations is the point $(2, 0)$ . This is the point where the line and the parabola intersect. Now, let's go back and check our answer choices to find the one that matches our solution.

We know that the solution is the point (2, 0). Examining the choices, we can directly identify the correct answer:
A. $(2,0)$ . This matches our solution exactly. The other options provide incorrect points. Therefore, we can definitively say that option A is the only correct answer. We have successfully solved the system of equations. Using this method, you can effectively solve various systems of equations.

Evaluating the Answer Choices: Eliminating the Impostors

Now that we have solved the system and found the solution $(2, 0)$ , let's take a look at the answer choices provided in the problem and explain why the other options are incorrect. This is an important step because it reinforces our understanding and helps us to avoid common mistakes.

A. (2, 0): This is the correct solution. We found that when $x = 2$ , $y = 0$ . This point satisfies both equations: $0 = 2 - 2$ (which is true) and $0 = 2^2 - 3(2) + 2$ (which is also true, as $0 = 4 - 6 + 2$ ).
B. (1, -1) and (3, 1): Let's test these points to see if they satisfy both equations. For $(1, -1)$ , we can substitute it into the first equation: $-1 = 1 - 2$ , which is true. Now, let's put it in the second equation: $-1 = 1^2 - 3(1) + 2$ , which simplifies to $-1 = 0$ , which is false. Therefore, $(1, -1)$ is not a solution. Now, let's try (3, 1). Putting it in the first equation: $1 = 3 - 2$ , which is true. In the second equation: $1 = 3^2 - 3(3) + 2$ , which simplifies to $1 = 2$ , which is false. Therefore, (3, 1) is not a solution. These points do not satisfy both equations simultaneously, indicating that they are incorrect solutions.
C. (2, 0) and (1, 0): We already know that $(2, 0)$ is a valid solution, but the point $(1, 0)$ can be eliminated by substituting it into the original equations. Substituting the point in the first equation: $0 = 1 - 2$ , this statement is false. The point does not satisfy the first equation. Thus, it cannot be a solution to the system.
D. (0, -2): Let's test this point. Substituting this in the first equation: $-2 = 0 - 2$ , which is true. Now, putting it in the second equation: $-2 = 0^2 - 3(0) + 2$ , which simplifies to $-2 = 2$ , which is false. This does not satisfy both equations. This point also does not satisfy both equations, so it's not a solution.

By carefully evaluating each option, we've confirmed that only option A is correct and the other options do not represent valid solutions for the given system of equations. This process of substitution and verification is crucial in understanding systems of equations.

Key Takeaways and Conclusion

Alright, folks, we've successfully navigated the world of systems of equations! Here's a quick recap of what we covered:

Understanding the Problem: Solving a system of equations means finding the point(s) where the equations intersect.
The Substitution Method: A powerful technique for solving systems, where we substitute one equation into the other to eliminate a variable.
Finding the Solution: We found the x-value, then used it to find the corresponding y-value, giving us the solution point (or points).
Verification: Always double-check your answer by plugging it back into the original equations to make sure it works!
Evaluating Answer Choices: We carefully assessed each option to confirm that only the correct solution satisfies both equations.

This method is super useful for all sorts of algebra problems. Keep practicing, and you'll become a pro in no time! Remember, the key is to break down the problem step-by-step and don't be afraid to double-check your work. You got this! Keep practicing, and you'll be solving these problems with ease. Until next time, happy solving!