Solving Systems Of Equations: First 3 Steps

Hey guys! Ever wondered how to solve a system of equations algebraically? It might seem daunting at first, but trust me, breaking it down into steps makes it super manageable. We're going to walk through the first three key steps using a concrete example. Let's dive in and make those equations less intimidating!

Step 1: Setting the Equations Equal to Each Other

The first step in solving a system of equations algebraically is to set the equations equal to each other. This works when both equations are already solved for the same variable, like 'y' in our example. Consider the system of equations:

y = x^2 - x - 3
y = -3x + 5

Since both equations are expressed in terms of y, we can set the right-hand sides equal to each other. This eliminates y and gives us a single equation in terms of x:

x^2 - x - 3 = -3x + 5

But why does this work, you might ask? Think of it this way: we're looking for the points where these two equations have the same y-value for a given x-value. By setting the equations equal, we're essentially finding the x-values where the graphs of these equations intersect. This intersection point(s) represents the solution(s) to the system.

This initial equation, x² - x - 3 = -3x + 5, is a quadratic equation. Solving it will give us the x-coordinates of the points of intersection. However, before we can solve for x, we need to rearrange the equation into a standard form that we can work with. This involves moving all the terms to one side, which leads us to the next crucial step.

Remember, this first step is critical. It transforms the problem from a system of two equations into a single equation that we can solve. Mastering this sets the foundation for the subsequent steps and brings us closer to finding the solution set. It's all about finding the common ground between the equations, the x-values where the y-values match up. So, make sure you've got this down before moving on. We're building a solid base here, guys!

Step 2: Rearranging the Equation into Standard Quadratic Form

Now that we've equated the two expressions, the second step is to rearrange the equation into the standard quadratic form. What's that, you ask? It's simply ax² + bx + c = 0, where a, b, and c are constants. Getting our equation into this form is essential because it allows us to use various methods like factoring, completing the square, or the quadratic formula to solve for x.

Let's take our equation from Step 1:

x^2 - x - 3 = -3x + 5

To rearrange it, we need to move all the terms to one side, leaving zero on the other side. We can do this by adding 3x to both sides and subtracting 5 from both sides:

x^2 - x - 3 + 3x - 5 = -3x + 5 + 3x - 5

Simplifying this gives us:

x^2 + 2x - 8 = 0

Voila! We now have a quadratic equation in the standard form ax² + bx + c = 0, where a = 1, b = 2, and c = -8. This transformation is a crucial step because it sets the stage for solving for x. By organizing the equation in this specific format, we unlock a whole toolbox of techniques that are designed to tackle quadratic equations. Think of it as translating a sentence into a language you understand – once it's in the right format, you can decipher its meaning.

Rearranging the equation is more than just a mathematical formality; it's a strategic move that simplifies the problem. It allows us to clearly identify the coefficients and apply the appropriate solving methods. So, make sure you’re comfortable with this rearrangement process. It's like prepping your ingredients before you start cooking – a necessary step to ensure a successful outcome. We're getting closer to finding those solutions, guys!

Step 3: Solving the Quadratic Equation for x

Alright, we've set the stage, and now comes the exciting part – the third step: solving the quadratic equation for x. We've got our equation in standard form: x² + 2x - 8 = 0. Now we need to find the values of x that make this equation true. There are a few ways we can do this, and the best method often depends on the specific equation. The most common methods are factoring, using the quadratic formula, and completing the square. For this example, factoring looks like the easiest route.

Factoring involves expressing the quadratic expression as a product of two binomials. We need to find two numbers that multiply to -8 (the constant term) and add up to 2 (the coefficient of the x term). After a little thought, we can see that the numbers 4 and -2 fit the bill, since 4 * -2 = -8 and 4 + (-2) = 2. So, we can factor the quadratic equation as follows:

(x + 4)(x - 2) = 0

Now, we use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This means that either (x + 4) = 0 or (x - 2) = 0. Solving each of these linear equations gives us our solutions for x:

x + 4 = 0  =>  x = -4
x - 2 = 0  =>  x = 2

So, we've found two x-values, -4 and 2, that satisfy the quadratic equation. These are the x-coordinates of the points where the two original equations intersect. But we're not quite done yet! Remember, we're solving a system of equations, so we need to find the corresponding y-values as well. Finding these x values is a huge step, though. You've basically unlocked half of the solution set!

Solving for x is the heart of the process. It's where the rubber meets the road, and you actually start to see the solutions emerge. Factoring is a powerful technique, but if it doesn't work, don't worry! The quadratic formula is always a reliable backup. The key is to choose the method that suits the equation best and to apply it carefully. We're on the home stretch now, guys, and you're doing great!

Next Steps: Finding the Corresponding y-values and the Solution Set

We've conquered the first three steps and found the x-values. Pat yourselves on the back! But the journey isn't over yet. The next crucial step involves finding the corresponding y-values for each of our x-values. Remember, we're looking for the points (x, y) that satisfy both equations in the original system. Once we have both the x and y coordinates, we can express the solution set, often written as a set of ordered pairs.

So, stay tuned, and let's complete this mission by finding those y-values and wrapping up the solution set! We're in this together, and you've got this!