Solving Equations Graphically: A Comprehensive Guide

Hey everyone, let's dive into a cool way to solve equations using graphs! We're gonna tackle the equation $-x-3 = x^2 - 2x - 15$ by graphing a system of equations. It's like a visual puzzle, and I promise, it's not as scary as it sounds. We will also determine the solutions of the equation using the graph we've created. Ready to get started? Let's go!

Understanding the Basics: Graphing for Solutions

Okay, so the main idea is this: When we graph a system of equations, the points where the lines or curves intersect are the solutions. Each intersection point represents an x-value that makes both equations true. In our case, we have a linear equation (a straight line) and a quadratic equation (a parabola – a U-shaped curve). So, the question asks us to use graphs to find the solutions for the equation $-x-3 = x^2 - 2x - 15$. But how do we do it? Easy! We'll represent each side of the equation as a separate equation in the system.

First, let's break down what's happening. The original equation is $-x-3 = x^2-2x-15$. To solve it graphically, we treat each side as a separate function. That is, we turn the equation into a system of two equations. The first equation, $y = -x - 3$, represents a straight line. The second, $y = x^2 - 2x - 15$, represents a parabola. The points where these two graphs intersect are the solutions to the original equation. Let's think step by step, the x-coordinate of the intersection points is our solution. So, our strategy is clear, we will graph these two functions and see where they meet.

Now, let's look at it more specifically. The original equation, $-x - 3 = x^2 - 2x - 15$, can be transformed into the system of equations you see in the problem, namely:

• $y = -x - 3$
• $y = x^2 - 2x - 15$

Each of these equations represents a function. The first one, $y = -x - 3$, is a linear function, which, when graphed, results in a straight line. The second, $y = x^2 - 2x - 15$, is a quadratic function, and its graph is a parabola. The points where the line and the parabola intersect are the solutions to the original equation.

So, why does this work? Basically, the x-values where the two graphs intersect are the values that make the expressions on both sides of the original equation equal. That's because at the point of intersection, the y-values (which represent the expressions $-x - 3$ and $x^2 - 2x - 15$) are the same.

To find the solution, we'll graph these two equations, $y = -x - 3$ and $y = x^2 - 2x - 15$, and find their points of intersection. The x-coordinates of these points are the solutions we're looking for! The graphical approach is a visual way to understand the solutions and is particularly helpful for visualizing the behavior of the equation.

Step-by-Step: Finding the Solutions Graphically

Alright, let's put our plan into action! Let's get down to the actual solving part. First off, we've got the equations:

• $y = -x - 3$ (This is a linear equation representing a line.)
• $y = x^2 - 2x - 15$ (This is a quadratic equation, graphing a parabola.)

To solve this, we need to graph both these equations and find where they intersect. Let's make it simpler, we will determine which option in the multiple-choice question is the solution for our equation. When it comes to linear equations, we should recall the slope-intercept form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Let's focus on the first equation, $y = -x - 3$. In this equation, the slope $m$ is -1, and the y-intercept $b$ is -3. This means that the line goes down 1 unit for every 1 unit it moves to the right, and it crosses the y-axis at the point (0, -3).

For the quadratic equation, $y = x^2 - 2x - 15$, we know it's a parabola. To graph this, we can find its vertex, which is the minimum or maximum point. The x-coordinate of the vertex of a parabola given by $y = ax^2 + bx + c$ is $-b / (2a)$. In our case, $a = 1$ and $b = -2$, so the x-coordinate is $-(-2) / (2*1) = 1$. Plug this x-value into the equation to find the y-coordinate: $y = (1)^2 - 2(1) - 15 = -16$. So, the vertex is at (1, -16). The parabola opens upwards because the coefficient of the $x^2$ term (which is 'a') is positive.

Now, imagine the two graphs on a coordinate plane. The line ($y = -x - 3$) and the parabola ($y = x^2 - 2x - 15$) will intersect at certain points. The x-coordinates of these intersection points are the solutions to the original equation $-x - 3 = x^2 - 2x - 15$. By carefully inspecting the graph, you would be able to pinpoint those x-values.

To find the solutions, we look for the points of intersection on the graph. Based on the multiple-choice options, we can check which of the provided x-values satisfy the original equation when plugged in. For this, it is necessary to manually check each option provided in the problem and see which of them are the correct answers. Let's consider the x-values provided in the multiple-choice options and determine the correct answer by substitution.

Finding the Solutions: Testing the Options

Okay, now that we've got the lowdown on the graphs, let's solve for those solutions! To find the x-values that satisfy the original equation, we're going to plug in each of the given options into the equations and check if they work. This is the simplest way to see which x-values actually are the intersection points.

Let's test the answer options one by one:

A. x = -7: Plug -7 into the original equation $-x - 3 = x^2 - 2x - 15$. This becomes $-(-7) - 3 = (-7)^2 - 2(-7) - 15$. Simplifying, we get $7 - 3 = 49 + 14 - 15$, which simplifies further to $4 = 48$. This is clearly not true, so -7 is not a solution.

B. x = -3: Substitute -3 into $-x - 3 = x^2 - 2x - 15$. This gives us $-(-3) - 3 = (-3)^2 - 2(-3) - 15$. Simplifying, we get $3 - 3 = 9 + 6 - 15$, which becomes $0 = 0$. This is true! So, -3 is a solution.

C. x = 0: Plugging in 0, we get $-(0) - 3 = (0)^2 - 2(0) - 15$, which becomes $-3 = -15$. This is not true, so 0 is not a solution.

D. x = 4: Substitute 4 into $-x - 3 = x^2 - 2x - 15$. This yields $-(4) - 3 = (4)^2 - 2(4) - 15$. Simplifying, we get $-7 = 16 - 8 - 15$, which results in $-7 = -7$. This is true! So, 4 is also a solution.

Therefore, we have found that x = -3 and x = 4 are the solutions.

Conclusion: The Final Answer

Alright, guys, we did it! We graphed our equations (in our minds, or perhaps on paper!), and by using the multiple-choice options, we were able to find the solution. Remember, the solutions to the equation $-x - 3 = x^2 - 2x - 15$ are the x-coordinates where the graphs of $y = -x - 3$ and $y = x^2 - 2x - 15$ intersect. By carefully checking the given options, we found that the solutions are -3 and 4.

So, the correct answers are B. -3 and D. 4. Easy peasy, right?

Graphing equations is a fantastic tool for visualizing and understanding solutions to equations, so keep practicing, and you'll be a pro in no time! Keep up the great work! That's all for today, folks. Thanks for hanging out and learning with me. I hope this explanation has helped you. If you have any questions, feel free to ask. Bye for now!