Solving Systems Of Equations: Substitution Method Example

Hey guys! Today, we're diving into the world of systems of equations and tackling them using the substitution method. This is a super useful technique in algebra, and we're going to break it down step-by-step so you can master it. We'll be working through a specific example to make things crystal clear. So, let's get started and solve this system together:

x + y = -10
-9x + 6y = 15

Understanding Systems of Equations

Before we jump into the substitution method, let's quickly recap what a system of equations actually is. Simply put, it's a set of two or more equations that share the same variables. Our goal is to find the values for these variables that satisfy all equations in the system simultaneously. Think of it as finding the sweet spot where all the equations agree.

There are several methods for solving these systems, including graphing, elimination, and, of course, the substitution method we're focusing on today. Each method has its strengths, and choosing the right one can make the solving process much easier. The substitution method shines when one equation can be easily solved for one variable in terms of the other, which is precisely the case in our example. So, let’s utilize this method.

Step 1: Solve One Equation for One Variable

The substitution method starts with isolating one variable in one of the equations. We want to get a variable by itself on one side of the equation. Looking at our system:

x + y = -10
-9x + 6y = 15

The first equation, x + y = -10, looks like the easier one to manipulate. Let's solve it for y. To do this, we simply subtract x from both sides of the equation:

y = -10 - x

Now we have y expressed in terms of x. This is a crucial step because we can now substitute this expression for y into the other equation. It’s all about finding clever ways to rewrite equations to make them simpler to work with, and this step is a perfect example of that.

Step 2: Substitute the Expression into the Other Equation

This is where the substitution method really gets its name! We're going to take the expression we found for y in the previous step (y = -10 - x) and substitute it into the second equation of our system, which is -9x + 6y = 15. Replacing y with (-10 - x) gives us:

-9x + 6(-10 - x) = 15

Notice what we've done here. We've effectively eliminated one variable (y) from the second equation. Now we have a single equation with only one variable (x), which we can solve. This is the power of substitution – it allows us to simplify the problem into something we can handle more easily. By reducing the complexity, we’re one step closer to finding our solution. Make sure to distribute carefully in the next step to avoid any errors.

Step 3: Solve for the Remaining Variable

Now we have the equation -9x + 6(-10 - x) = 15. Let's solve for x. First, we need to distribute the 6:

-9x - 60 - 6x = 15

Next, combine like terms:

-15x - 60 = 15

Now, add 60 to both sides:

-15x = 75

Finally, divide both sides by -15:

x = -5

So, we've found the value of x! It's -5. This is a major milestone in solving our system. But remember, we're not done yet. We still need to find the value of y. This is where our earlier work pays off, as we already have an expression for y in terms of x. We're on the home stretch now!

Step 4: Substitute the Value Back to Find the Other Variable

We've found that x = -5. Now we need to find y. Remember in Step 1, we solved the first equation for y and got y = -10 - x. We can now substitute the value of x we just found into this equation:

y = -10 - (-5)

Simplify:

y = -10 + 5
y = -5

So, y = -5 as well! We've now found the values for both x and y that satisfy the system of equations. This is a fantastic achievement! But to be absolutely sure we have the correct solution, it’s always a good idea to check our work.

Step 5: Check Your Solution

To check our solution, we need to substitute the values we found for x and y (which are both -5) back into the original equations. This will ensure that our solution works for both equations in the system.

Let's start with the first equation, x + y = -10:

(-5) + (-5) = -10
-10 = -10

This checks out! Now let's try the second equation, -9x + 6y = 15:

-9(-5) + 6(-5) = 15
45 - 30 = 15
15 = 15

This also checks out! Since our solution satisfies both equations, we can confidently say that we've found the correct answer. It’s so satisfying when everything lines up perfectly, isn’t it?

Solution

The solution to the system of equations is:

x = -5
y = -5

We can also write this as an ordered pair: (-5, -5). This represents the point where the two lines represented by the equations intersect on a graph. Understanding the graphical representation can give you an even deeper understanding of what we've just done algebraically.

Key Takeaways

• The substitution method is a powerful tool for solving systems of equations.
• The key is to isolate one variable in one equation and substitute its expression into the other equation.
• Always check your solution by plugging the values back into the original equations.
• Remember to take your time and work carefully to avoid mistakes, especially when distributing and combining like terms.

Practice Makes Perfect

The best way to master the substitution method is to practice! Work through various examples, and you'll become more comfortable with the steps involved. Don't be afraid to make mistakes – they're a natural part of the learning process. Each mistake is an opportunity to learn and improve. So, grab some practice problems and keep honing your skills. You got this!

Wrapping Up

And there you have it! We've successfully solved a system of equations using the substitution method. I hope this step-by-step guide has been helpful. Remember, mathematics is like building blocks; each concept builds upon the previous one. Mastering systems of equations is a valuable skill that will help you in more advanced math courses. Keep practicing, stay curious, and you'll be solving even more complex problems in no time. Keep up the great work, guys!