Solving Systems Of Equations Substitution And Elimination Methods

Hey guys! Let's dive into the fascinating world of solving systems of equations. It's like being a detective, piecing together clues to find the values of unknown variables. In this article, we'll tackle a specific system of equations, but more importantly, we'll explore various methods to solve these kinds of problems. Get ready to sharpen your mathematical skills!

Understanding Systems of Equations

Before we jump into the nitty-gritty, let's make sure we're all on the same page. A system of equations is simply a set of two or more equations that share the same variables. Our goal? To find the values of those variables that satisfy all equations in the system simultaneously. Think of it as finding the sweet spot that works for every equation.

In our case, we have the following system:

4x - 5y = 11
3x - 6y = 6

This is a system of two linear equations with two variables, x and y. Now, there are several ways we can crack this code, and we'll explore them in detail.

Why are Systems of Equations Important?

You might be wondering, "Why should I care about solving systems of equations?" Well, these systems pop up in all sorts of real-world scenarios. From calculating the break-even point for a business to determining the trajectory of a rocket, systems of equations are essential tools in mathematics, science, engineering, and economics. Mastering these techniques opens doors to solving a wide range of practical problems. So, let's get started!

Methods for Solving Systems of Equations

Okay, let's get to the fun part – the actual solving! There are several methods at our disposal, each with its own strengths and weaknesses. We'll focus on the most common and effective ones:

1. Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation. It's like replacing a piece in a puzzle to simplify the picture.
2. Elimination Method (or Addition Method): This method focuses on eliminating one variable by adding or subtracting multiples of the equations. It's like strategically canceling out terms to isolate the variable you want.
3. Graphical Method: This method involves graphing the equations and finding the point(s) where the lines intersect. It's a visual approach that can be super helpful for understanding the solutions.
4. Matrix Methods: For larger systems of equations, matrix methods like Gaussian elimination and matrix inversion can be very efficient. We won't delve too deeply into these here, but it's good to know they exist.

For our specific system, we'll primarily focus on the substitution and elimination methods, as they are particularly well-suited for this type of problem. However, understanding the graphical method provides a great visual check for our solutions.

Solving the System Using the Substitution Method

Let's start with the substitution method. Remember, the key here is to isolate one variable in one equation and then substitute that expression into the other equation.

Step 1: Choose an equation and solve for one variable.

Looking at our system:

4x - 5y = 11
3x - 6y = 6

The second equation, 3x - 6y = 6, looks a bit simpler to work with. Let's solve it for x:

3x = 6y + 6
x = (6y + 6) / 3
x = 2y + 2

Great! We've isolated x in terms of y. Now, the magic happens.

Step 2: Substitute the expression into the other equation.

We'll substitute our expression for x (2y + 2) into the first equation, 4x - 5y = 11:

4(2y + 2) - 5y = 11

See what we did? We replaced x with its equivalent expression in terms of y. Now we have an equation with only one variable, y, which we can solve.

Step 3: Solve the resulting equation.

Let's simplify and solve for y:

8y + 8 - 5y = 11
3y + 8 = 11
3y = 3
y = 1

Awesome! We found that y = 1. We're halfway there!

Step 4: Substitute the value back into either equation to find the other variable.

Now that we know y = 1, we can plug it back into any of our equations to find x. Let's use the expression we found earlier, x = 2y + 2:

x = 2(1) + 2
x = 2 + 2
x = 4

We've got it! x = 4.

Step 5: Check your solution.

It's always a good idea to check our solution by plugging the values of x and y back into the original equations:

For 4x - 5y = 11:

4(4) - 5(1) = 16 - 5 = 11  (Correct!)

For 3x - 6y = 6:

3(4) - 6(1) = 12 - 6 = 6  (Correct!)

Our solution checks out! We've successfully solved the system using the substitution method.

Therefore, the solution to the system of equations is x = 4 and y = 1.

Cracking the Code Using the Elimination Method

Now, let's tackle the same system using the elimination method. This approach focuses on eliminating one variable by manipulating the equations.

Step 1: Multiply one or both equations by a constant so that the coefficients of one variable are opposites.

Looking at our system:

4x - 5y = 11
3x - 6y = 6

We want to make the coefficients of either x or y opposites. Let's target the x coefficients. To do this, we can multiply the first equation by 3 and the second equation by -4:

3 * (4x - 5y) = 3 * 11  =>  12x - 15y = 33
-4 * (3x - 6y) = -4 * 6  =>  -12x + 24y = -24

Notice that the coefficients of x are now 12 and -12 – perfect opposites!

Step 2: Add the equations together.

Now, we add the two modified equations together. This will eliminate the x variable:

(12x - 15y) + (-12x + 24y) = 33 + (-24)
9y = 9

The x terms canceled out, leaving us with a simple equation in terms of y.

Step 3: Solve for the remaining variable.

Let's solve for y:

9y = 9
y = 1

Just like before, we found that y = 1.

Step 4: Substitute the value back into either original equation to find the other variable.

We substitute y = 1 back into either of the original equations. Let's use the first equation, 4x - 5y = 11:

4x - 5(1) = 11
4x - 5 = 11
4x = 16
x = 4

We found that x = 4, which matches our result from the substitution method.

Step 5: Check your solution.

As before, we check our solution by plugging the values of x and y back into the original equations. We already did this in the substitution method section, and we know it works!

Therefore, the solution to the system of equations using the elimination method is also x = 4 and y = 1.

A Quick Look at the Graphical Method

While we've solved our system algebraically, let's briefly consider the graphical method. Each equation in our system represents a line. The solution to the system is the point where these lines intersect.

If you were to graph the equations:

4x - 5y = 11
3x - 6y = 6

You would find that they intersect at the point (4, 1), which visually confirms our solution of x = 4 and y = 1. This method provides a great visual check and can be particularly useful for understanding systems of inequalities.

Dependent Equations and Infinite Solutions

Now, let's touch on a special case: dependent equations. Sometimes, when solving a system, you might encounter a situation where the equations are essentially multiples of each other. This means they represent the same line, and there are infinitely many solutions.

In such cases, you won't get a unique solution like x = 4 and y = 1. Instead, you'll end up with an identity (like 0 = 0) when trying to solve the system. To express the solution, you write one variable in terms of the other.

For example, if you had a system where the equations simplified to x + y = 2 and 2x + 2y = 4, you'd recognize that the second equation is just a multiple of the first. The solution would be expressed as something like x = 2 - y, indicating that for any value of y, there's a corresponding value of x that satisfies the system.

Thankfully, our original system wasn't dependent. We got a unique solution, which is the most common scenario.

Conclusion Mastering the Art of Solving Equations

Alright, guys! We've successfully tackled a system of equations using both the substitution and elimination methods. We also touched on the graphical method and the concept of dependent equations. By understanding these techniques, you've equipped yourself with valuable tools for solving a wide range of mathematical problems.

Remember, practice makes perfect! The more you work with systems of equations, the more comfortable and confident you'll become. So, keep those pencils sharp and your minds even sharper. Happy solving!