Solving Systems Of Equations By Substitution A Step By Step Guide

Hey guys! Are you struggling with systems of equations? Don't worry, you're not alone! One of the most powerful techniques for solving these problems is the method of substitution. In this article, we're going to dive deep into this method, breaking it down step-by-step with a clear example. We'll explore the underlying concepts, provide a detailed walkthrough of the solution, and even discuss some helpful tips and tricks to master this essential mathematical skill.

What are Systems of Equations?

Before we jump into the nitty-gritty, let's quickly recap what systems of equations actually are. Simply put, a system of equations is a set of two or more equations that share the same variables. Our goal is to find the values of these variables that satisfy all equations in the system simultaneously. Think of it like finding a secret code that unlocks all the equations at once!

These systems pop up all over the place, from simple algebra problems to complex real-world scenarios. Whether you're calculating the break-even point for your lemonade stand or designing a bridge, understanding how to solve systems of equations is a crucial skill.

The Substitution Method: A Step-by-Step Approach

The substitution method is a clever technique that allows us to solve for the variables one at a time. The basic idea is to isolate one variable in one of the equations and then substitute that expression into the other equation. This eliminates one variable, leaving us with a single equation that we can easily solve. Once we find 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 break this down into manageable steps:

1. Isolate a Variable: Choose one of the equations and solve it for one of the variables. This means getting one variable all by itself on one side of the equation. Look for an equation where a variable already has a coefficient of 1 or -1, as this will make the isolation process easier.
2. Substitute: Take the expression you found in step 1 and substitute it into the other equation. This means replacing the variable in the second equation with the entire expression you isolated in the first equation. This will result in a new equation with only one variable.
3. Solve: Solve the new equation for the remaining variable. This is usually a straightforward algebraic process.
4. Back-Substitute: Once you've found the value of one variable, plug it back into either of the original equations (or the expression you isolated in step 1) to solve for the other variable.
5. Check: To ensure your solution is correct, substitute the values you found for both variables into both original equations. If both equations are satisfied, you've found the correct solution!

Example Time: Let's Solve a System!

Okay, let's put this into practice with the system you provided:

$ egin{aligned} 3x - 5y &= 11 \ y &= 2x - 5 ind{aligned} $

This system is perfect for demonstrating the substitution method. Notice that the second equation, y = 2x - 5, already has y isolated. This makes our job much easier!

Step 1: Isolate a Variable (Already Done!)

As we just mentioned, the second equation is already solved for y. We have y = 2x - 5. Fantastic!

Step 2: Substitute

Now, we'll substitute this expression for y into the first equation:

$ 3x - 5(2x - 5) = 11 $

See what we did there? We replaced the y in the first equation with the entire expression 2x - 5. This is the heart of the substitution method! We've eliminated y from the first equation, leaving us with an equation that only involves x.

Step 3: Solve

Now, let's solve this equation for x. First, we need to distribute the -5:

$ 3x - 10x + 25 = 11 $

Next, combine like terms:

$ -7x + 25 = 11 $

Subtract 25 from both sides:

$ -7x = -14 $

Finally, divide both sides by -7:

$ x = 2 $

Woohoo! We've found the value of x! x equals 2.

Step 4: Back-Substitute

Now that we know x = 2, we can plug this value back into either of the original equations to solve for y. Let's use the second equation, y = 2x - 5, as it's already solved for y:

$ y = 2(2) - 5 $

Simplify:

$ y = 4 - 5 $

$ y = -1 $

Awesome! We've found the value of y: y equals -1.

Step 5: Check

To be absolutely sure our solution is correct, let's plug both x = 2 and y = -1 into both original equations:

• Equation 1: $ 3x - 5y = 11 $$ 3(2) - 5(-1) = 6 + 5 = 11 $ This equation is satisfied!
• Equation 2: $ y = 2x - 5 $$ -1 = 2(2) - 5 = 4 - 5 = -1 $ This equation is also satisfied!

Since both equations are satisfied, we've confirmed that our solution is correct.

The Solution

Therefore, the solution to the system of equations is:

$ x = 2 $$ y = -1 $

We can also write this as an ordered pair: (2, -1).

Tips and Tricks for Mastering Substitution

• Choose Wisely: When deciding which equation to use for isolating a variable, look for the equation where a variable has a coefficient of 1 or -1. This will minimize fractions and make the algebra cleaner.
• Distribute Carefully: When substituting an expression into another equation, make sure to distribute any coefficients correctly. This is a common source of errors.
• Double-Check Your Work: Always check your solution by plugging the values of x and y back into both original equations. This will help you catch any mistakes.
• Practice Makes Perfect: The more you practice the substitution method, the more comfortable you'll become with it. Work through various examples, and you'll soon be solving systems of equations like a pro!

When Substitution Shines (and When It Doesn't)

The substitution method is a fantastic tool, but it's not always the best choice for every system of equations. It's particularly effective when:

• One of the equations is already solved for a variable.
• It's easy to isolate a variable in one of the equations.

However, if neither equation has a variable with a coefficient of 1 or -1, and it would be messy to isolate a variable, the elimination method might be a better option. The elimination method involves adding or subtracting the equations to eliminate one of the variables.

Conclusion: You've Got This!

The substitution method is a powerful technique for solving systems of equations. By following the steps outlined in this article, you can confidently tackle these problems. Remember to isolate a variable, substitute the expression, solve for the remaining variable, back-substitute to find the other variable, and always check your solution. With practice and perseverance, you'll master this essential mathematical skill. Keep up the great work, guys!