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

Hey Plastik Magazine readers! Let's dive into a classic math problem: finding the solutions to a system of equations. Specifically, we're tackling the following system:

\left\{\begin{array}{r} x^2+y^2=25 \\ 2 x+y=-5 \\end{array}\right.

This might look a bit intimidating at first, but trust me, we'll break it down into manageable steps. The key is to understand what we're actually trying to find: the points where the two equations intersect on a graph. The first equation, $x^2 + y^2 = 25$, represents a circle centered at the origin (0, 0) with a radius of 5. The second equation, $2x + y = -5$, represents a straight line. Therefore, the solutions to the system are the points where the line crosses the circle. Let's get started, guys!

Step-by-Step Solution

Isolating a Variable

Our initial step in solving the system of equations is to isolate a variable in one of the equations. The second equation, $2x + y = -5$, is perfect for this because it's linear and easy to manipulate. We can isolate y by subtracting $2x$ from both sides:

$y = -2x - 5$

Now, we have an expression for y in terms of x. This is gold because it allows us to substitute this value into the first equation.

Substitution is Key

Now, we'll substitute the expression for y (which is $-2x - 5$) into the first equation, $x^2 + y^2 = 25$. This will give us a quadratic equation in terms of x:

$x^2 + (-2x - 5)^2 = 25$

Let's expand and simplify this equation. Remember to square the binomial correctly:

$x^2 + (4x^2 + 20x + 25) = 25$

Combining like terms, we get:

$5x^2 + 20x + 25 = 25$

Subtracting 25 from both sides, we further simplify to:

$5x^2 + 20x = 0$

Solving for x

Great, we've got a quadratic equation! We can solve this by factoring. Notice that both terms have a common factor of $5x$. Factoring this out, we get:

$5x(x + 4) = 0$

For this equation to be true, either $5x = 0$ or $(x + 4) = 0$. This gives us two possible solutions for x:

• $x = 0$
• $x = -4$

Finding the Corresponding y Values

We've found our x values; now it's time to find the corresponding y values. We'll use the equation we derived earlier, $y = -2x - 5$, to do this.

• When $x = 0$:

  $y = -2(0) - 5 = -5$. So, one solution is $(0, -5)$.

• When $x = -4$:

  $y = -2(-4) - 5 = 8 - 5 = 3$. So, another solution is $(-4, 3)$.

Final Answer

Therefore, the solutions to the system of equations are $(0, -5)$ and $(-4, 3)$.

Analyzing the Answer Choices

Now that we've solved the system, let's look at the answer choices provided. Remember, we're looking for the pairs of (x, y) values that satisfy both equations. Let's assess each option:

• A. (0, -5) and (-5, 5): We found that (0, -5) is a solution. Let's check if (-5, 5) fits. Substituting these values into the first equation, we get $(-5)^2 + 5^2 = 25 + 25 = 50$, which is not equal to 25. Thus, this option is incorrect.

• B. (0, -5) and (5, -15): We know (0, -5) is a solution. Let's check (5, -15). Substituting into the first equation, we get $5^2 + (-15)^2 = 25 + 225 = 250$, not 25. This option is also out.

• C. (0, -5) and (-4, 3): We found that these two points are the solutions! We checked earlier, and (0, -5) works. Let's double-check (-4, 3): $(-4)^2 + 3^2 = 16 + 9 = 25$ and $2(-4) + 3 = -8 + 3 = -5$. Both equations check out. This is the correct answer.

• D. (0, -5) and (4, -13): (0, -5) works. Let's see about (4, -13): $4^2 + (-13)^2 = 16 + 169 = 185$, not 25. Incorrect.

Therefore, the correct answer is C.

Visualizing the Solution

It's always helpful to visualize what's going on, guys. If you were to graph the circle and the line, you'd see that they intersect at the points (0, -5) and (-4, 3). This graphical representation confirms our algebraic solution and helps build a deeper understanding of the problem.

Tips and Tricks for Solving Systems of Equations

To become a master of solving systems of equations, keep these tips in mind:

• Practice, practice, practice: The more problems you solve, the more comfortable you'll become with the different methods.
• Be organized: Write down each step clearly, so you can easily follow your work and identify any errors.
• Double-check your answers: Always substitute your solutions back into the original equations to make sure they are correct. This is super important to avoid silly mistakes.
• Understand the concepts: Make sure you understand the underlying concepts, like the properties of circles and lines. This will help you choose the best approach for each problem.
• Use technology wisely: Calculators and online tools can be helpful for checking your answers and visualizing the graphs, but make sure you understand the manual methods first.

Wrapping Up

So there you have it, our friends! We've successfully solved a system of equations by using substitution, factoring, and checking our answers. Remember that practice is key, and with time, you'll become a pro at these types of problems. Keep exploring, keep learning, and as always, stay curious! We hope you enjoyed the explanation. Let us know what other math topics you'd like us to cover in the comments. Until next time!