Solving Systems Of Equations By Substitution A Step By Step Guide

Hey guys! Let's dive into the world of solving systems of equations using the substitution method. It might sound intimidating, but trust me, it's a super useful tool in your math arsenal. We're going to break it down step-by-step, so you'll be a substitution pro in no time. We'll tackle a specific example, but the principles we cover will apply to all sorts of systems of equations.

What are Systems of Equations?

Before we jump into the substitution method, let's quickly recap what systems of equations are all about. Simply put, a system of equations is a set of two or more equations that share the same variables. Our goal? To find the values of those variables that make all the equations in the system true at the same time. Think of it like a puzzle where you need to find the perfect pieces that fit together.

For example, the system we'll be working with today is:

x - y = 4
3x + 2y = -28

Here, we have two equations, and both of them involve the variables 'x' and 'y'. Our mission, should we choose to accept it, is to find the values of 'x' and 'y' that satisfy both equations simultaneously. There are several methods to solve systems of equations, and today we're focusing on the substitution method, which is particularly handy when one of the equations can be easily rearranged to isolate a variable.

The Substitution Method: A Step-by-Step Breakdown

The substitution method is all about replacing one variable in an equation with an equivalent expression involving the other variable. This might sound a bit abstract, but it becomes crystal clear when we walk through the steps. Here's the general roadmap:

1. Solve one equation for one variable: Choose one of the equations in the system and solve it for one of its variables. This means isolating that variable on one side of the equation. Look for equations where a variable has a coefficient of 1 or -1, as these are usually the easiest to work with. This step is crucial as it sets the stage for the substitution process. By isolating one variable, we create an expression that represents its value in terms of the other variable, which we can then use to substitute into the other equation.
2. Substitute the expression into the other equation: Take the expression you found in step 1 and substitute it into the other equation in the system. This means replacing the variable you solved for with the entire expression. The result will be a new equation that contains only one variable. This is where the magic happens! By substituting, we eliminate one of the variables, transforming the system into a single equation that we can solve directly. This step is the heart of the substitution method, as it allows us to reduce the complexity of the system.
3. Solve the new equation: Solve the equation you obtained in step 2 for the remaining variable. This should be a standard algebraic equation that you can solve using familiar techniques. Once you've solved for one variable, you're halfway there! You've found one piece of the puzzle, and now you just need to find the other.
4. Substitute the value back to find the other variable: Once you know the value of one variable, substitute it back into either of the original equations (or the expression you found in step 1) to solve for the other variable. This step completes the solution process. By substituting the value we found back into one of the equations, we can easily solve for the remaining variable, giving us the complete solution to the system.
5. Check your solution: It's always a good idea to check your solution by substituting the values of both variables back into both original equations. If both equations are true, then you've found the correct solution. This final step is essential to ensure accuracy. By plugging our solution back into the original equations, we can verify that it satisfies the entire system, giving us confidence in our answer.

Let's Solve an Example: x - y = 4 and 3x + 2y = -28

Okay, let's put these steps into action with the system we mentioned earlier:

x - y = 4
3x + 2y = -28

Step 1: Solve one equation for one variable

Looking at our two equations, x - y = 4 seems like a good candidate to start with because the coefficient of 'x' is 1. Let's solve this equation for 'x':

x - y = 4
x = y + 4  (Add 'y' to both sides)

Great! We've isolated 'x' and now we have an expression for it: x = y + 4. This expression tells us that 'x' is equal to 'y' plus 4. We'll use this in the next step.

Step 2: Substitute the expression into the other equation

Now, we'll take our expression for 'x' (x = y + 4) and substitute it into the other equation, which is 3x + 2y = -28. This means we'll replace 'x' in the second equation with the entire expression (y + 4):

3x + 2y = -28
3(y + 4) + 2y = -28  (Substitute 'x' with 'y + 4')

See what happened? We replaced 'x' with (y + 4), and now we have an equation that only involves the variable 'y'. This is exactly what we wanted!

Step 3: Solve the new equation

Let's simplify and solve the equation we just obtained:

3(y + 4) + 2y = -28
3y + 12 + 2y = -28  (Distribute the 3)
5y + 12 = -28  (Combine like terms)
5y = -40  (Subtract 12 from both sides)
y = -8  (Divide both sides by 5)

Awesome! We've found the value of 'y': y = -8. That's one variable down, one to go!

Step 4: Substitute the value back to find the other variable

Now that we know y = -8, we can substitute this value back into either of the original equations to solve for 'x'. However, it's often easiest to substitute it back into the expression we found in step 1, which is x = y + 4:

x = y + 4
x = -8 + 4  (Substitute 'y' with -8)
x = -4

Excellent! We've found the value of 'x': x = -4. So, our solution is x = -4 and y = -8.

Step 5: Check your solution

To be absolutely sure, let's check our solution by substituting x = -4 and y = -8 back into both original equations:

• Equation 1: x - y = 4

  -4 - (-8) = 4
  -4 + 8 = 4
  4 = 4  (True)
  

• Equation 2: 3x + 2y = -28

  3(-4) + 2(-8) = -28
  -12 - 16 = -28
  -28 = -28  (True)
  

Both equations are true! This confirms that our solution x = -4 and y = -8 is correct.

Key Takeaways and Tips for Success

The substitution method is a powerful tool for solving systems of equations, but here are a few key takeaways and tips to keep in mind:

• Choose wisely: When deciding which equation to solve for which variable in step 1, look for the easiest option. Equations with variables that have coefficients of 1 or -1 are usually the best choice.
• Be careful with signs: Pay close attention to signs (positive and negative) throughout the process, especially when substituting and distributing. A small sign error can throw off your entire solution.
• Double-check your work: Always check your solution by substituting the values back into the original equations. This is the best way to catch any mistakes and ensure accuracy.
• Practice makes perfect: Like any math skill, the more you practice the substitution method, the more comfortable and confident you'll become. Work through various examples to solidify your understanding.

Substitution Method vs. Other Methods

You might be wondering, "When should I use the substitution method, and when should I use other methods like elimination?" That's a great question!

The substitution method shines when one of the equations can be easily solved for one variable. If you can quickly isolate a variable, substitution is often a good choice. On the other hand, the elimination method (also known as the addition method) is often preferred when the coefficients of one of the variables are opposites or can be easily made opposites by multiplying one or both equations by a constant.

Ultimately, the best method depends on the specific system of equations you're dealing with. With practice, you'll develop a knack for recognizing which method is most efficient for a given problem.

Let's Practice!

Now that you've got the hang of the substitution method, it's time to put your skills to the test! Try solving these systems of equations using substitution:

1. 2x + y = 7 x - y = 2
2. 4x - 3y = 10 x + 2y = 1
3. y = 3x - 5 6x - 2y = 10

Remember to follow the steps we discussed, and don't forget to check your solutions!

Conclusion

The substitution method is a valuable tool for solving systems of equations. By mastering this method, you'll be able to tackle a wide range of mathematical problems. Keep practicing, and you'll become a system-solving superstar! Remember, the key is to break down the problem into smaller, manageable steps and to double-check your work along the way. Happy solving, guys!