Solve Equations 2x2 By Substitution Method
Hey guys! Today, we're diving into the fascinating world of solving systems of equations, specifically focusing on the substitution method. If you've ever felt a bit lost when faced with two equations and two unknowns (like x and y), don't worry, you're in the right place. We're going to break it down step by step, making it super clear and easy to understand. Let's use the example you provided: 4x + 2y = 60 and 3x + 2y = 48. Trust me, by the end of this article, you'll be solving these like a pro!
Understanding Systems of Equations
Before we jump into the nitty-gritty, let’s quickly recap what a system of equations actually is. Think of it as 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 true simultaneously. In our case, we want to find the values of x and y that satisfy both 4x + 2y = 60 and 3x + 2y = 48. There are several methods to tackle these problems, and today, we're mastering the substitution method. This method is particularly handy when one of the variables can be easily isolated. Now, why is this important? Well, systems of equations pop up all over the place in real-world scenarios, from balancing chemical equations to planning a budget. So, understanding how to solve them is a seriously valuable skill.
Why the Substitution Method?
The substitution method is a powerful technique for solving systems of equations. What makes it so special? Well, it’s all about simplifying the problem by expressing one variable in terms of the other. This way, we can reduce a two-variable equation into a single-variable equation, which is much easier to solve. It’s like turning a complex puzzle into a simpler one! This method shines when one of the equations has a variable with a coefficient of 1 or -1. This makes isolating that variable straightforward. But even if the coefficients aren't that friendly, the substitution method can still be applied with a little extra algebraic maneuvering. The key is to choose the easiest variable to isolate, minimizing fractions and complex steps. By mastering this method, you'll have a solid tool in your math arsenal to tackle a wide range of problems.
Step-by-Step Guide to the Substitution Method
Alright, let's get down to business and walk through the steps of solving our system of equations using the substitution method. Remember, we're working with 4x + 2y = 60 and 3x + 2y = 48. Grab your pencil and paper, and let's do this!
Step 1: Choose an Equation and Isolate a Variable
The first move in the substitution game is to pick one of the equations and isolate one of the variables. This means getting a variable all by itself on one side of the equation. The goal here is to choose the equation and variable that look easiest to work with. Looking at our equations, let’s choose the second equation, 3x + 2y = 48, and decide to isolate y. Why y? Well, it doesn't really matter in this case since both equations have a '2y' term, making it a straightforward choice. To isolate y, we'll first subtract 3x from both sides of the equation:
3x + 2y - 3x = 48 - 3x
This simplifies to:
2y = 48 - 3x
Now, to get y completely alone, we divide both sides by 2:
2y / 2 = (48 - 3x) / 2
Which gives us:
y = 24 - (3/2)x
Great! We've successfully isolated y in terms of x. This is a crucial step, so make sure you're comfortable with the algebra involved. We now have an expression for y that we can use in the next step.
Step 2: Substitute the Expression into the Other Equation
Now comes the substitution part of the substitution method! We have an expression for y from the previous step: y = 24 - (3/2)x. We're going to take this expression and plug it into the other equation – the one we didn't use in Step 1. That’s 4x + 2y = 60. Wherever we see a y in this equation, we'll replace it with 24 - (3/2)x. So, the equation becomes:
4x + 2(24 - (3/2)x) = 60
Notice how we've replaced y with our expression. Now we have an equation with only one variable, x, which is something we can solve! This step is all about reducing the problem to a single variable, making it much easier to handle. Take your time here and double-check your substitution to avoid any errors. The rest of the solution hinges on this step, so let's make sure we get it right.
Step 3: Solve for the Remaining Variable
We've successfully substituted, and now we have the equation 4x + 2(24 - (3/2)x) = 60. The next step is to solve for x. This involves simplifying the equation and isolating x on one side. First, let's distribute the 2 across the terms inside the parentheses:
4x + 48 - 3x = 60
Now, combine the x terms:
x + 48 = 60
To isolate x, subtract 48 from both sides:
x + 48 - 48 = 60 - 48
This simplifies to:
x = 12
Woohoo! We've found the value of x. This is a significant milestone in solving the system of equations. Remember, the key here was careful simplification and algebraic manipulation. We're halfway there – now we just need to find the value of y.
Step 4: Substitute the Solved Variable Back to Find the Other
Alright, we've cracked the code for x; we know that x = 12. Now, to find y, we need to substitute this value back into one of our original equations or the expression we found for y in Step 1. Let's use the expression we derived earlier, y = 24 - (3/2)x, as it's already set up to solve for y. Plug in x = 12:
y = 24 - (3/2)(12)
Now, let's simplify. First, multiply (3/2) by 12:
y = 24 - 18
Finally, subtract:
y = 6
Fantastic! We've found that y = 6. This step is all about using the value we found for one variable to unlock the value of the other. By substituting back, we complete the puzzle and find the solution to the system of equations.
Step 5: Check Your Solution
Before we celebrate, it’s crucial to make sure our solution is correct. We do this by plugging the values we found for x and y back into both of the original equations. If our solution is correct, it should make both equations true. Let's start with the first equation, 4x + 2y = 60. Substitute x = 12 and y = 6:
4(12) + 2(6) = 60
48 + 12 = 60
60 = 60
Great! It checks out for the first equation. Now, let’s try the second equation, 3x + 2y = 48:
3(12) + 2(6) = 48
36 + 12 = 48
48 = 48
Perfect! It works for the second equation as well. Since our values satisfy both equations, we can confidently say that our solution is correct. Checking your solution is a vital step in problem-solving, as it helps you catch any mistakes and ensures you get the right answer. It’s like the final seal of approval on your hard work!
Solution and Conclusion
So, after all that awesome work, we've arrived at the solution! We found that x = 12 and y = 6. This means that the point (12, 6) is the intersection of the two lines represented by our equations. In other words, it's the one and only pair of values that makes both equations true at the same time. Give yourselves a pat on the back – you've successfully solved a system of equations using the substitution method!
Wrapping Up
To wrap things up, let's quickly recap the key steps we took: We started by isolating a variable in one equation, then we substituted that expression into the other equation. This gave us a single-variable equation, which we solved. Finally, we substituted the value we found back into one of the equations to find the value of the other variable, and we checked our solution. The substitution method is a powerful tool for tackling systems of equations, and with practice, you'll become a master at it. Remember, the key is to take it one step at a time, be careful with your algebra, and always check your work. You've got this!
I hope this guide has made solving systems of equations using the substitution method crystal clear for you guys. Keep practicing, and you'll be solving these problems with confidence in no time! If you have any questions or want to try another example, just let me know. Happy solving!