Solving Systems Of Equations: Substitution Method

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Hey math enthusiasts! Today, we're diving into the substitution method for solving systems of equations. Systems of equations pop up all over the place, from figuring out the best phone plan to predicting the path of a rocket. Understanding how to solve them is a super useful skill. We'll walk through a specific example, step-by-step, making sure you grasp the concepts. So, grab your pencils and let's get started!

Understanding the Basics: Systems of Equations

Alright, before we jump into the example, let's quickly recap what a system of equations actually is. Basically, it's a set of two or more equations, and we're trying to find the values of the variables (usually x and y) that satisfy all the equations in the system simultaneously. Think of it like a puzzle where you need to find the pieces that fit perfectly in multiple places. The solution to a system of equations is the point (or points) where the lines represented by the equations intersect on a graph. This point gives us the values of x and y that work in both equations. There are several ways to solve these systems, and today, we're focusing on the substitution method. It's a fantastic technique when one of the equations is already solved for a variable, which, as you'll see, makes things much easier. Systems of equations can represent many real-world problems. For instance, imagine you're comparing two different gym memberships. One might have a lower monthly fee but a higher per-visit charge, while the other is the opposite. To figure out which membership is cheaper, depending on how often you go, you’d set up a system of equations. The variables would represent the monthly fee and the cost per visit, and the solution would tell you how many visits would make the costs equal. Beyond personal finance, systems of equations are used everywhere! They are used in computer graphics to calculate the position of objects in 3D space, or in economics to model supply and demand. Even in engineering, they help design circuits and structures! Understanding and mastering them unlocks a world of problem-solving possibilities. This substitution method is a core skill! Keep an eye on the details, and you’ll be solving all sorts of problems in no time! So, with the basics covered, let’s dig into how the substitution method works.

Let's Solve: The Substitution Method in Action

Okay, time for the main event! Let's solve this system of equations using the substitution method:

  • Equation 1: 4x - 4y = 28
  • Equation 2: x = -2y - 11

Notice how Equation 2 is already solved for x. This makes substitution super easy! Here’s how it works, step by step:

Step 1: Substitution Time!

Since we know that x is equal to -2y - 11, we can substitute this expression for x in Equation 1. So, wherever we see x in Equation 1, we replace it with -2y - 11. This gives us:

4(-2y - 11) - 4y = 28

See? We've successfully eliminated x from the equation, and now we only have y to solve for. That's the beauty of substitution – it reduces the problem to a single variable!

Step 2: Simplify and Solve for y

Now, let's simplify and solve for y. First, distribute the 4 across the terms inside the parentheses:

-8y - 44 - 4y = 28

Next, combine like terms:

-12y - 44 = 28

Then, add 44 to both sides of the equation:

-12y = 72

Finally, divide both sides by -12:

y = -6

Boom! We've found the value of y! y equals -6. You're making great progress!

Step 3: Solve for x

Now that we know y = -6, we can plug this value back into either of the original equations to solve for x. But, because Equation 2 is already solved for x, it's the easiest choice. Let’s substitute y = -6 into Equation 2:

x = -2y - 11 x = -2(-6) - 11 x = 12 - 11 x = 1

Awesome! We've found that x = 1.

Step 4: The Solution

We've found our x and our y. That means we have successfully solved the system of equations. We can write our solution as an ordered pair (x, y), which in this case is:

(1, -6)

This means that the point (1, -6) is the point where the two lines represented by the equations intersect on a graph. This is the only point that satisfies both equations simultaneously. Congrats, you've solved your first system of equations using substitution!

Checking Your Work: A Crucial Step

Always, always check your solution! Plugging your x and y values back into the original equations is an important habit. It ensures that your solution is correct. Let's do that for our solution (1, -6):

Check in Equation 1

4x - 4y = 28 4(1) - 4*(-6) = 28* 4 + 24 = 28 28 = 28 (This checks out!)

Check in Equation 2

x = -2y - 11 1 = -2(-6) - 11* 1 = 12 - 11 1 = 1 (This also checks out!)

Since our solution works in both original equations, we know we've found the correct answer. Checking your work helps catch any silly mistakes and gives you confidence in your solution.

More Examples for Practice

Want some more practice? Here are a couple more systems of equations for you to try solving using the substitution method. Remember to follow the steps we covered, and always check your answers!

  1. egin{cases}y = x + 3 \ 2x + y = 9 egin{cases}

  2. egin{cases}3x - y = 7 \ x = 2y - 1 egin{cases}

Go ahead and work through these. The answers are provided in the next section, so you can check your work.

Answers to Practice Problems

  1. (3, 6)
  2. (3, 1)

Keep practicing, and you'll become a substitution pro in no time! Remember the main steps – substitute, simplify, solve, and always check your answers. Math can be fun when you have a solid strategy!

Tips and Tricks for Success

To really master the substitution method, here are a few tips and tricks:

  • Choose the easiest equation: If one of the equations is already solved for a variable, use it! It'll save you a ton of time and effort.
  • Be careful with signs: Pay close attention to positive and negative signs when substituting and simplifying. A small mistake here can lead to the wrong answer.
  • Simplify carefully: Double-check your arithmetic when simplifying the equations. It's easy to make a small error, so take your time.
  • Practice, practice, practice: The more you practice, the more comfortable you'll become with the method. Work through plenty of examples to build your confidence.
  • Don't be afraid to rearrange: If neither equation is already solved for a variable, rearrange one of them to isolate a variable. This will set you up for successful substitution.
  • Check your work religiously: As mentioned before, always substitute your solution back into the original equations to make sure it's correct. This habit will save you from making silly mistakes and boost your confidence.

By keeping these tips in mind, you will be well on your way to becoming a substitution master! The substitution method is a versatile tool. It’s useful not only in math classes but also in everyday problem-solving scenarios. Take a little time to master it, and the rest will follow. There are other methods for solving systems of equations, such as elimination and graphing. These methods also have their own strengths, but the substitution method is a great place to begin.

Conclusion: You've Got This!

Alright, folks, you've reached the end! We've covered the basics of systems of equations, the step-by-step process of the substitution method, and even offered some practice problems. Remember, the key is to understand the concept and practice regularly. Don't worry if it takes a little time to grasp it fully. Math is like any other skill – the more you practice, the better you get. Keep practicing, keep learning, and don't be afraid to ask for help if you need it. You've got this! Happy solving!