Solving Systems Of Equations Find The Smallest X Coordinate

by Sharif Sakr 60 views

Hey guys! Today, we're diving into the exciting world of solving systems of equations. Specifically, we're going to tackle a system involving a parabola and a line, and our ultimate goal is to find the smallest x-coordinate where these two graphs intersect. Trust me, it's not as daunting as it sounds! We'll break it down step by step, making sure everyone's on board. So, grab your thinking caps, and let's get started!

Understanding Systems of Equations

Before we jump into the nitty-gritty, let's take a moment to understand what a system of equations actually is. In simple terms, a system of equations is a set of two or more equations that share the same variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. Graphically, the solution represents the point(s) where the graphs of the equations intersect. Think of it like finding the common ground between two different relationships.

In our case, we have two equations:

  1. y = x2 - 5
  2. y - x = 1

The first equation represents a parabola, which is a U-shaped curve. The second equation represents a straight line. Our mission, should we choose to accept it (and we do!), is to find the point(s) where this parabola and line cross paths. This means finding the x and y values that satisfy both equations. There are several methods to do this, but we'll focus on the substitution method today.

The Substitution Method: A Step-by-Step Approach

The substitution method is a powerful technique for solving systems of equations. The basic idea is to solve one equation for one variable and then substitute that expression into the other equation. This eliminates one variable, leaving us with a single equation in one variable, which we can then solve using standard algebraic techniques. Once we've found the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.

Let's apply this to our system. Looking at the second equation, y - x = 1, it seems relatively easy to solve for y. We can simply add x to both sides to get:

y = x + 1

Now, we have an expression for y in terms of x. This is where the substitution magic happens! We'll substitute this expression for y into the first equation:

(x + 1) = x2 - 5

See what we did there? We replaced y in the first equation with (x + 1), effectively eliminating y from the equation. Now we have a quadratic equation in x, which we can solve.

Solving the Quadratic Equation

Our new equation is x + 1 = x2 - 5. To solve this quadratic equation, we first need to rearrange it into standard form, which is ax2 + bx + c = 0. Let's subtract (x + 1) from both sides:

0 = x2 - x - 6

Now we have a quadratic equation in standard form. There are several ways to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. For this particular equation, factoring seems like the easiest approach. We need to find two numbers that multiply to -6 and add to -1. Those numbers are -3 and 2. So, we can factor the quadratic as:

0 = (x - 3)(x + 2)

According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we have two possible solutions for x:

x - 3 = 0 or x + 2 = 0

Solving these equations gives us:

x = 3 or x = -2

So, we have two x-coordinates where the parabola and line intersect: 3 and -2. Remember, the problem asks for the smallest x-coordinate, so our answer is -2.

Finding the Corresponding y-Coordinates

We're not quite done yet! We've found the x-coordinates of the intersection points, but we also need to find the corresponding y-coordinates. To do this, we can substitute each x-value back into either of the original equations. Let's use the simpler equation, y = x + 1:

For x = 3:

y = 3 + 1 = 4

For x = -2:

y = -2 + 1 = -1

So, the two intersection points are (3, 4) and (-2, -1). This confirms that our smallest x-coordinate is -2.

Graphical Interpretation

It's always a good idea to visualize what we've done. If you were to graph the parabola y = x2 - 5 and the line y = x + 1, you would see that they intersect at the points (3, 4) and (-2, -1). The point (-2, -1) is to the left of (3, 4), confirming that -2 is indeed the smallest x-coordinate.

Common Mistakes to Avoid

Solving systems of equations can be tricky, so let's talk about some common mistakes to watch out for:

  • Forgetting to substitute back: Once you've solved for one variable, don't forget to substitute it back into one of the original equations to find the value of the other variable. This is a crucial step in finding the complete solution.
  • Making algebraic errors: Be careful with your algebra! A small mistake in rearranging equations or factoring can lead to incorrect answers. Double-check your work, especially when dealing with negative signs.
  • Not checking your solutions: After you've found the solution(s), plug them back into the original equations to make sure they work. This is a great way to catch any errors you might have made.
  • Misinterpreting the question: Always read the question carefully to understand what it's asking for. In this case, we were asked for the smallest x-coordinate, so we needed to make sure we selected the correct value from our solutions.

Practice Makes Perfect

The best way to master solving systems of equations is to practice, practice, practice! The more problems you solve, the more comfortable you'll become with the different techniques and the less likely you'll be to make mistakes. So, grab some practice problems, put your skills to the test, and remember, you've got this!

Conclusion: Mastering the Art of Solving Systems

And there you have it! We've successfully solved a system of equations involving a parabola and a line, and we've found the smallest x-coordinate of the intersection points. We used the substitution method, factored a quadratic equation, and even talked about common mistakes to avoid. Remember, solving systems of equations is a fundamental skill in algebra and calculus, so it's well worth the effort to master it. Keep practicing, keep learning, and you'll be solving systems of equations like a pro in no time! So remember guys, if you face a similar problem, always remember the steps we took: Substitution, simplification, and solving for the unknown. You got this!