Is It A Solution? Checking Ordered Pairs In Equations

Hey Plastik Magazine readers! Let's dive into something that might seem a bit tricky at first, but trust me, it's totally manageable: figuring out if a given ordered pair is a solution to a system of equations. This concept is super important in math, and it's like a building block for more complex stuff. So, let's break it down step by step and make sure you've got this locked in. Think of it like this: you're given a set of instructions (the equations), and you're testing if a specific location (the ordered pair) follows those instructions. If it does, boom, it's a solution. If not, then it doesn't fit the mold. We're going to take the ordered pair (-3, 4) and the system of equations: 2/3x - 3/2y = -8. The goal is to figure out if this ordered pair makes both equations true. If it does, then it's a solution to the system. If it doesn't, then it's not a solution. Ready to jump in? Let's go!

Understanding Ordered Pairs and Systems of Equations

Okay, before we get our hands dirty with the calculations, let's make sure we're all on the same page about what an ordered pair and a system of equations are. An ordered pair is just a pair of numbers, like (-3, 4), and the order matters! The first number in the pair always represents the x-coordinate (horizontal position), and the second number represents the y-coordinate (vertical position). So, in our case, -3 is the x-coordinate, and 4 is the y-coordinate. This pair represents a single point on a graph. A system of equations is simply a set of two or more equations that we're trying to solve together. Each equation in the system usually represents a line when graphed. The solution to a system of equations is the point (or points) where all the lines intersect. In other words, it's the point(s) that satisfy all the equations in the system. But how do we determine if an ordered pair is part of the solution? We substitute the values of the ordered pair into the equations and check if they are true. If they are, that's your solution, but if not, then it is not.

The Importance of Correct Substitution

One of the most common mistakes people make when solving these problems is making calculation errors during substitution. So, let's go slow and steady, taking it one step at a time. The first equation we have to work with is: 2/3x - 3/2y = -8. This is the perfect time to recall that x is -3 and y is 4 from the ordered pair (-3, 4). Now, let's substitute those values into the equation. Wherever you see 'x,' replace it with -3, and wherever you see 'y,' replace it with 4. When we do this, the equation becomes (2/3)*(-3) - (3/2)*(4) = -8. Now it's a simple matter of doing the math and verifying the results. Remember to follow the order of operations, so multiplication and division come before addition and subtraction. In this case, we have a mix of multiplication and subtraction to deal with.

Step-by-Step: Testing the Ordered Pair

Alright, let's get into the nitty-gritty of checking if our ordered pair (-3, 4) is a solution to the given system of equations. Here is the first equation to check: 2/3x - 3/2y = -8. As stated, the x-value is -3, and the y-value is 4. First, let's substitute the values into the equation: (2/3)*(-3) - (3/2)*(4) = -8. Now, perform the calculations. When you multiply 2/3 by -3, you get -2. And when you multiply 3/2 by 4, you get 6. Therefore, our equation now looks like this: -2 - 6 = -8. Simplify the left side by doing the subtraction, and you get -8 = -8. Since -8 equals -8, the first equation holds true for the ordered pair (-3, 4). It is a good start, but remember that for an ordered pair to be considered a solution to the system, it needs to satisfy all equations in the system. If it doesn't satisfy every equation, the ordered pair is not a solution. Keep this in mind when you are working with multiple equations.

Checking for Mistakes and Double-Checking

When you're dealing with numbers and equations, it's easy to make a small mistake that throws everything off. That's why it's super important to double-check your work. After substituting the values and doing the math, take a moment to look back at each step. Did you substitute the values correctly? Did you remember the order of operations? Did you copy down the numbers and signs correctly? One of the best ways to check your work is to do the calculations twice. The first time, go through the steps as usual. The second time, work backward. This might seem like a waste of time, but it can save you from a lot of frustration down the road. Another great way to check your work is to ask a friend or classmate to look over your work and see if they can catch any mistakes. They might spot something you missed. Also, don't be afraid to use a calculator. It can be a great tool to help you with the calculations, especially when dealing with fractions and decimals. Just make sure you understand how to do the calculations by hand too, in case you don't have a calculator handy.

Conclusion: Is (-3,4) a Solution?

So, guys, is the ordered pair (-3, 4) a solution to the system? Well, let's recap. We started with the ordered pair (-3, 4) and the equation 2/3x - 3/2y = -8. We substituted -3 for x and 4 for y into the equation, and we ended up with -8 = -8, which is a true statement. Therefore, because we only have one equation to test, and the ordered pair (-3, 4) works, we can confidently say that (-3, 4) is a solution to the equation 2/3x - 3/2y = -8. If there were a system of equations (more than one equation), we would have had to make sure (-3, 4) worked in all of the equations in that system for it to be considered a solution to the system. But, since we only had one to test, we are good to go! Math can seem complex, but it can also be very straightforward when you break it down into smaller steps. Keep practicing, and you will become a pro at this. Keep learning, keep exploring, and keep rocking your math journey!