Solving System Of Equations: Y = X² + X - 2 And Y = 2x - 2
Hey guys! Let's dive into solving a system of equations where we have a quadratic equation and a linear equation. Specifically, we're going to figure out the solution(s) for the graphed system of equations y = x² + x - 2 and y = 2x - 2. This means we need to find the points where these two graphs intersect. Understanding how to solve these systems is super important in algebra and it pops up in various real-world applications too. So, grab your thinking caps, and let's get started!
Understanding the Equations
Before we jump into solving, let’s break down what these equations represent. First, we have y = x² + x - 2. This is a quadratic equation, and when graphed, it forms a parabola. The general form of a quadratic equation is y = ax² + bx + c, and in our case, a is 1, b is 1, and c is -2. The shape of the parabola depends on the coefficients, and understanding these coefficients helps us visualize the graph.
Next up, we have y = 2x - 2. This is a linear equation, which means it forms a straight line when graphed. The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. In our equation, the slope m is 2, and the y-intercept b is -2. Knowing the slope and y-intercept makes it easy to draw this line on a graph. To effectively visualize the solutions, you'll want to understand how each equation behaves graphically.
When we talk about solving a system of equations, we're essentially looking for the point(s) where these two graphs meet. These points of intersection represent the (x, y) values that satisfy both equations simultaneously. Think of it as finding the common ground between the parabola and the straight line. This concept is crucial because it’s not just about finding numbers that work for one equation but finding the numbers that make both equations true at the same time. This gives us the solutions to the system.
Methods to Solve the System of Equations
Alright, let's talk about the cool ways we can solve this system. There are mainly two methods we can use: the substitution method and the graphical method. Each method has its own charm, and depending on the equations, one might be easier than the other. Let’s explore both so you can pick your favorite or use the one that fits best for the situation.
1. Substitution Method
The substitution method is a super handy algebraic technique. The main idea is to solve one equation for one variable and then substitute that expression into the other equation. This way, we reduce the system to a single equation with a single variable, which is much easier to solve. Once we find the value of one variable, we can plug it back into either of the original equations to find the value of the other variable. Let's see how it works step-by-step with our equations.
First, we notice that both equations are already solved for y. This makes our job easier! We have y = x² + x - 2 and y = 2x - 2. Since both expressions are equal to y, we can set them equal to each other:
x² + x - 2 = 2x - 2
Now, we have a quadratic equation in terms of x. Let’s rearrange it to the standard form ax² + bx + c = 0 by subtracting 2x and adding 2 to both sides:
x² + x - 2 - 2x + 2 = 0
x² - x = 0
Next, we can factor out an x from the equation:
x(x - 1) = 0
This gives us two possible solutions for x: x = 0 and x - 1 = 0, which means x = 0 and x = 1. Now that we have the x values, we can find the corresponding y values by plugging them back into either of the original equations. The linear equation y = 2x - 2 looks simpler, so let’s use that one.
For x = 0:
y = 2(0) - 2 = -2
So, one solution is (0, -2).
For x = 1:
y = 2(1) - 2 = 0
So, the other solution is (1, 0).
Therefore, the solutions to the system of equations using the substitution method are (0, -2) and (1, 0). This method is awesome because it's precise and gives us the exact solutions without relying on graphs.
2. Graphical Method
The graphical method is a visual way to solve the system of equations. It involves plotting both equations on the same coordinate plane and identifying the points where the graphs intersect. These intersection points are the solutions to the system because they satisfy both equations simultaneously. It’s like finding where the two equations agree on a map. This method is especially helpful for understanding the nature of the solutions visually.
Let's plot the two equations: y = x² + x - 2 and y = 2x - 2. To plot the quadratic equation, y = x² + x - 2, we can find some key points. The parabola opens upwards since the coefficient of x² is positive. We can also find the x-intercepts by setting y = 0 and solving for x:
0 = x² + x - 2
0 = (x + 2)(x - 1)
So, the x-intercepts are x = -2 and x = 1. The y-intercept is found by setting x = 0:
y = (0)² + (0) - 2 = -2
So, the y-intercept is (0, -2). We can also find the vertex of the parabola using the formula x = -b / 2a, where a = 1 and b = 1:
x = -1 / (2 * 1) = -0.5
Plugging x = -0.5 into the equation gives us the y-coordinate of the vertex:
y = (-0.5)² + (-0.5) - 2 = 0.25 - 0.5 - 2 = -2.25
So, the vertex is (-0.5, -2.25). With these key points, we can sketch the parabola.
Now, let's plot the linear equation y = 2x - 2. We already know the slope is 2 and the y-intercept is -2. We can also find another point by plugging in a value for x, say x = 1:
y = 2(1) - 2 = 0
So, another point on the line is (1, 0). With the y-intercept and this point, we can draw the line.
When you plot both equations on the same graph, you’ll see that they intersect at two points. By visually inspecting the graph, we can estimate the coordinates of these points. If we plot the graphs accurately, we'll see that the intersection points are (0, -2) and (1, 0). These are the same solutions we found using the substitution method!
The graphical method is awesome because it gives you a visual understanding of the solutions. It’s like seeing the problem and the solution in one picture. However, keep in mind that the accuracy of this method depends on how precisely you draw the graphs. For exact solutions, algebraic methods like substitution are generally more reliable.
Step-by-Step Solution Using Substitution
Let's walk through the substitution method again, step-by-step, just to make sure we've got it down. Sometimes seeing it more than once helps solidify the understanding. We’ll use the same equations, y = x² + x - 2 and y = 2x - 2, and break it down into easy-to-follow steps.
Step 1: Set the equations equal to each other
Since both equations are already solved for y, we can set them equal to each other. This is the key first step in the substitution method when both equations are in the form y = ...:
x² + x - 2 = 2x - 2
Step 2: Rearrange the equation to the standard quadratic form
We want to get all the terms on one side to set the equation to zero. This puts it in the standard quadratic form ax² + bx + c = 0, which is easier to solve. Subtract 2x from both sides and add 2 to both sides:
x² + x - 2 - 2x + 2 = 0
x² - x = 0
Step 3: Factor the quadratic equation
Now, we factor the quadratic equation. In this case, we can factor out an x from both terms:
x(x - 1) = 0
Step 4: Solve for x
To find the solutions for x, we set each factor equal to zero:
x = 0 or x - 1 = 0
Solving for x in each case gives us:
x = 0 and x = 1
Step 5: Substitute the x values back into one of the original equations to find the corresponding y values
We have two x values, so we need to find the y values that correspond to each. We can use either of the original equations, but the linear equation y = 2x - 2 is simpler. For x = 0:
y = 2(0) - 2 = -2
So, one solution is (0, -2).
For x = 1:
y = 2(1) - 2 = 0
So, the other solution is (1, 0).
Step 6: State the solutions
We found two solutions for the system of equations. These are the points where the parabola and the line intersect:
(0, -2) and (1, 0)
And there you have it! We’ve walked through the substitution method step-by-step, making it super clear how to find the solutions to this system of equations. Practice makes perfect, so try this method with other systems too!
Verification of the Solutions
To be absolutely sure that we’ve got the correct solutions, it’s a great idea to verify them. This means plugging our solutions back into the original equations to see if they hold true. It’s like double-checking your work to make sure everything adds up. Let’s verify our solutions (0, -2) and (1, 0) for the system of equations y = x² + x - 2 and y = 2x - 2.
Verifying the solution (0, -2)
First, we'll plug x = 0 and y = -2 into the quadratic equation y = x² + x - 2:
-2 = (0)² + (0) - 2
-2 = -2
This is true, so the solution (0, -2) satisfies the first equation.
Now, let's plug the same values into the linear equation y = 2x - 2:
-2 = 2(0) - 2
-2 = -2
This is also true, so the solution (0, -2) satisfies the second equation as well. Since it satisfies both equations, it’s definitely a solution to the system.
Verifying the solution (1, 0)
Next, we'll plug x = 1 and y = 0 into the quadratic equation y = x² + x - 2:
0 = (1)² + (1) - 2
0 = 1 + 1 - 2
0 = 0
This is true, so the solution (1, 0) satisfies the first equation.
Now, let's plug the same values into the linear equation y = 2x - 2:
0 = 2(1) - 2
0 = 2 - 2
0 = 0
This is also true, so the solution (1, 0) satisfies the second equation. Since it satisfies both equations, it’s a valid solution to the system.
By verifying both solutions in both equations, we can be super confident that we’ve found the correct answers. This step is crucial because it catches any potential errors and ensures that our solutions are accurate. Always remember to verify your solutions whenever you can – it’s a great habit to develop!
Conclusion
Alright, guys, we've successfully navigated through solving the system of equations y = x² + x - 2 and y = 2x - 2! We've explored two powerful methods: the substitution method and the graphical method. The substitution method gave us the precise solutions (0, -2) and (1, 0) through algebraic manipulation, while the graphical method allowed us to visualize these solutions as the intersection points of the parabola and the line. We even took the extra step to verify our solutions, ensuring they hold true for both equations.
Understanding how to solve systems of equations is super valuable in mathematics. It's not just about finding the right numbers; it's about understanding how different equations interact and relate to each other. These skills will come in handy in more advanced math topics and real-world applications too. Keep practicing, and you’ll become a pro at solving these systems in no time! Keep up the awesome work!