Finding The Intersection Point: Solving Systems Of Equations

Hey math whizzes! Today, we're diving deep into the awesome world of solving systems of equations to find that elusive point of intersection. You know, when those lines or planes just have to meet? We're talking about a specific set of equations that, when solved together, reveal that exact spot where they all converge. It's like finding the X on a treasure map, guys!

So, what exactly is a system of equations? Basically, it's a collection of two or more equations that share the same set of variables. When we're asked to solve a system, we're on a mission to find values for those variables that make all the equations in the system true simultaneously. And when we're looking for the point of intersection, we're specifically interested in that one unique solution (or sometimes multiple solutions, but let's stick to the simpler case for now!) that satisfies every single equation.

Think about it visually. For linear equations in two variables (like y = mx + b), each equation represents a line on a graph. The point where those lines cross is the solution to the system. When we move to three variables, we're talking about planes in 3D space. The intersection point is where these planes slice through each other, meeting at a single coordinate (x, y, z). Pretty neat, right?

There are several cool methods we can use to crack these systems. We've got substitution, elimination, and even matrix methods like Gaussian elimination or Cramer's rule. Each has its own flavor, and sometimes one is way easier than the other depending on the specific equations you're wrestling with. The key is to pick a method, stick with it, and do the algebra carefully. One tiny slip-up, and you might end up with a totally wrong intersection point!

Let's get our hands dirty with a specific example that really showcases how we find this intersection point. We're going to tackle this beast:

$$\left\{\begin{array}{c} 3 x-y+z=-10 \\ 2 x-4 y-z=20 \\ 6 x+8 y+z=-100 \end{array}\right.$$

See all those variables and numbers? Don't let it scare you! This is where the magic happens. We're looking for that single (x, y, z) triplet that makes all three of these statements true at the same time. This is the heart of finding the point of intersection for these three planes.

The Elimination Method: Our Go-To Strategy

For this particular system, the elimination method seems like a fantastic choice. Why? Because look at the z terms! We have +z in the first and third equations, and -z in the second. This is an algebraic gift, making it super easy to eliminate z by adding equations together. Let's make this our primary strategy to find the intersection point.

Step 1: Eliminate z from two pairs of equations.

Our first target is to get rid of z using two different combinations of the original equations. This will leave us with two new equations, each containing only x and y. This is a crucial step in reducing the complexity of finding the intersection point.

• Pair 1: Equation 1 and Equation 2 Let's add Equation 1 and Equation 2 directly. Notice how the +z and -z will cancel each other out:

  (3x - y + z) + (2x - 4y - z) = -10 + 20
  5x - 5y = 10
  

  Boom! That was easy. We've got our first equation with just x and y. Let's call this Equation 4.

• Pair 2: Equation 2 and Equation 3 Now, let's use Equation 2 and Equation 3. We need to get rid of z here too. Since we have -z in Equation 2 and +z in Equation 3, we can add them directly:

  (2x - 4y - z) + (6x + 8y + z) = 20 + (-100)
  8x + 4y = -80
  

  Awesome! We've got our second equation with just x and y. Let's call this Equation 5.

Now, we've successfully reduced our system of three equations with three variables down to a system of two equations with two variables. This is a huge leap towards finding our intersection point!

Solving the New System for x and y

We now have a simpler system:

$$\left\{\begin{array}{c} 5 x-5 y=10 \quad \text{(Equation 4)} \\ 8 x+4 y=-80 \quad \text{(Equation 5)} \end{array}\right.$$

Our goal is to find the x and y values that satisfy both of these equations. From there, we can easily plug those values back into one of the original equations to find z. This will give us our final intersection point!

We can use either substitution or elimination again. Let's stick with elimination because it often feels more direct when you have coefficients that are easy to work with.

Step 2: Eliminate either x or y from the new system (Equations 4 and 5).

To make things neat, let's simplify Equation 4 first by dividing everything by 5:

x - y = 2

And let's simplify Equation 5 by dividing everything by 4:

2x + y = -20

Look at this! We now have y with a coefficient of -1 in the first simplified equation and +1 in the second. This is perfect for elimination! We can add these two simplified equations together to eliminate y.

• Add the simplified equations:

  (x - y) + (2x + y) = 2 + (-20)
  3x = -18
  

  Now, we can easily solve for x:

  x = -18 / 3
  x = -6
  

  We found our first coordinate for the intersection point! Only two more to go.

Step 3: Substitute the value of x back into one of the simplified equations to find y.

Let's use the simplified x - y = 2 equation. We know x = -6, so:

-6 - y = 2

Now, let's solve for y:

-y = 2 + 6
-y = 8
y = -8

Fantastic! We've now found the y-coordinate of our intersection point. We're so close to cracking this case!

Finding the Final Piece: The z-coordinate

We have x = -6 and y = -8. The final step to pinpoint our intersection point is to substitute these values back into any of the original three equations to solve for z. Let's pick the first equation, 3x - y + z = -10, because it looks pretty straightforward.

Step 4: Substitute the values of x and y into one of the original equations to find z.

Using Equation 1: 3x - y + z = -10

Substitute x = -6 and y = -8:

3(-6) - (-8) + z = -10
-18 + 8 + z = -10
-10 + z = -10

Now, solve for z:

z = -10 + 10
z = 0

And there you have it, folks! We've found all three coordinates: x = -6, y = -8, and z = 0.

The Grand Reveal: The Point of Intersection

The point of intersection for the given system of equations is (-6, -8, 0). This single point satisfies all three original equations, meaning the three planes represented by these equations all meet at this exact location in 3D space.

Verification (Always a Good Idea!)

To be absolutely sure, let's plug these values back into the other two original equations to confirm they hold true. This is your check to ensure you've nailed the intersection point!

• Equation 2: 2x - 4y - z = 20 2(-6) - 4(-8) - 0 = -12 - (-32) - 0 = -12 + 32 = 20. Correct!

• Equation 3: 6x + 8y + z = -100 6(-6) + 8(-8) + 0 = -36 + (-64) + 0 = -36 - 64 = -100. Correct!

See? Both equations hold true. This confirms that (-6, -8, 0) is indeed the correct point of intersection.

Why Is This So Important?

Understanding how to solve systems of equations and find intersection points is fundamental in so many areas of math and science. Whether you're dealing with physics problems involving forces, engineering challenges with structural integrity, economics modeling market equilibrium, or even computer graphics rendering complex scenes, finding where different conditions or constraints meet is often the key to solving the problem. It's the core concept of finding a state where multiple requirements are simultaneously satisfied. So, next time you see a system of equations, remember you're not just crunching numbers; you're locating a critical meeting point!

Keep practicing, guys, and you'll become masters at finding these intersection points. Happy solving!