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

Hey guys! Today, we're diving into a fun math problem: finding a solution to a system of equations. Specifically, we're tackling the following system:

$$\begin{cases} y - 3 = x \\ x^2 - 6x + 13 = y \end{cases}$$

And we need to figure out which of the following options is a valid solution:

A. $(-5, 2)$
B. $(-2, 1)$
C. $(2, 5)$
D. $(8, 5)$

Let's break this down step by step to make sure we understand exactly what's going on and how to solve it.

Understanding the Problem

Before we start crunching numbers, it's important to understand what a "solution" to a system of equations actually means. A solution is a set of values (in this case, an x and a y) that makes both equations true at the same time. Think of it like this: it's a secret code that unlocks both equations simultaneously. Our job is to find that code among the given options.

Why is this important? Because simply plugging the numbers into one equation isn't enough. A solution must satisfy all equations in the system. This is a fundamental concept in algebra, and mastering it opens the door to solving more complex problems later on.

Visualizing the Equations

Another helpful way to think about this is graphically. Each equation represents a curve (in this case, a line and a parabola) on a coordinate plane. The solutions to the system are the points where these curves intersect. So, we're essentially looking for the coordinates of the intersection point.

Strategy Time!

Now that we understand the problem, let's talk strategy. The most straightforward way to solve this is to substitute one equation into the other. Since we have y - 3 = x, we can easily express y in terms of x and then substitute that expression into the second equation. This will give us a single equation with only x, which we can then solve. Once we find the value(s) of x, we can plug them back into either of the original equations to find the corresponding value(s) of y. Finally, we'll check which of the given options match our solution.

Solving the System

Okay, let's get our hands dirty with some algebra!

Step 1: Express y in terms of x

From the first equation, y - 3 = x, we can easily isolate y:
y = x + 3

Step 2: Substitute into the second equation

Now, we'll substitute this expression for y into the second equation, x^2 - 6x + 13 = y:

x^2 - 6x + 13 = x + 3

Step 3: Simplify and solve for x

Let's rearrange the equation to get a quadratic equation in standard form:

x^2 - 6x - x + 13 - 3 = 0

x^2 - 7x + 10 = 0

Now we need to factor this quadratic. We're looking for two numbers that multiply to 10 and add up to -7. Those numbers are -2 and -5. So, we can factor the quadratic as follows:

(x - 2)(x - 5) = 0

This gives us two possible solutions for x:

x = 2 or x = 5

Step 4: Find the corresponding y values

Now that we have the values of x, we can plug them back into the equation y = x + 3 to find the corresponding values of y:

• If x = 2, then y = 2 + 3 = 5
• If x = 5, then y = 5 + 3 = 8

So, our solutions are (2, 5) and (5, 8).

Checking the Options

Alright, we found our solutions! Now, let's see which of the given options matches our results:

A. (-5, 2) - No match. B. (-2, 1) - No match. C. (2, 5) - Match! D. (8, 5) - No match.

Therefore, the correct answer is C. (2, 5).

Why Other Options are Wrong

It's helpful to understand why the other options are incorrect. Let's take a quick look:

• A. (-5, 2): If we substitute x = -5 into y = x + 3, we get y = -5 + 3 = -2, not 2. So, this point doesn't satisfy the first equation.
• B. (-2, 1): If we substitute x = -2 into y = x + 3, we get y = -2 + 3 = 1, which satisfies the first equation. However, if we substitute x = -2 and y = 1 into the second equation, we get (-2)^2 - 6(-2) + 13 = 4 + 12 + 13 = 29, which is not equal to 1. So, this point doesn't satisfy the second equation.
• D. (8, 5): If we substitute x = 8 into y = x + 3, we get y = 8 + 3 = 11, not 5. So, this point doesn't satisfy the first equation.

Key Takeaway: Always make sure your solution satisfies both equations in the system!

Alternative Method: Testing the Options

Another approach, especially useful when you have multiple-choice options, is to directly test each option in both equations. This can sometimes be faster than solving the entire system from scratch.

Let's try it with option C (2, 5):

• Equation 1: y - 3 = x Substituting x = 2 and y = 5, we get 5 - 3 = 2, which is true.
• Equation 2: x^2 - 6x + 13 = y Substituting x = 2 and y = 5, we get 2^2 - 6(2) + 13 = 4 - 12 + 13 = 5, which is also true.

Since option C satisfies both equations, it's the correct solution. This method can be a good way to quickly check your work or to solve the problem if you're short on time.

Tips and Tricks for Solving Systems of Equations

Solving systems of equations is a crucial skill in algebra and beyond. Here are a few tips and tricks to help you master it:

• Substitution is your friend: Look for opportunities to substitute one equation into another to simplify the problem. This is especially useful when one of the equations is already solved for one of the variables.
• Elimination is also powerful: If you have two equations with the same coefficients for one of the variables, you can eliminate that variable by adding or subtracting the equations. This is particularly handy when dealing with linear equations.
• Check your work: Always, always, always check your solutions by plugging them back into the original equations. This will help you catch any errors and ensure that your solution is correct.
• Practice makes perfect: The more you practice solving systems of equations, the better you'll become at recognizing patterns and choosing the most efficient solution methods. Don't be afraid to try different approaches and learn from your mistakes.
• Understand the graphical interpretation: Visualizing the equations as curves on a coordinate plane can give you a deeper understanding of what a solution represents. It can also help you identify potential solutions and check your work.

Conclusion

So, there you have it! We successfully solved the system of equations and found that the correct answer is C. (2, 5). Remember the key steps: substitute, simplify, solve, and check! Keep practicing, and you'll become a system-solving pro in no time. Keep an eye out for more math adventures, and happy solving!