Are (1, 0) And (3, 12/7) Solutions To -6x + 7y = -6? A Step-by-Step Guide

Hey everyone! Today, we're diving into a bit of math to figure out if some ordered pairs are solutions to a specific equation. Specifically, we're looking at the equation -6x + 7y = -6, and we want to know if the ordered pairs (1, 0) and (3, 12/7) fit the bill. Think of it like this: we're detectives, and the equation is our crime scene. The ordered pairs are our suspects, and we need to see if they match the evidence. So, grab your magnifying glasses (or, you know, your calculators), and let's get started!

Understanding Ordered Pairs and Equations

Before we jump into the calculations, let's quickly recap what ordered pairs and equations are all about. An ordered pair, like (1, 0) or (3, 12/7), is simply a set of two numbers written in a specific order. The first number, usually called 'x', represents the horizontal position on a graph, and the second number, 'y', represents the vertical position. Think of it as coordinates on a map! An equation, on the other hand, is a mathematical statement that shows the relationship between variables (like 'x' and 'y') and constants (like -6 and 7). Our equation, -6x + 7y = -6, tells us how 'x' and 'y' are related to each other. To determine if an ordered pair is a solution to an equation, we need to see if the equation holds true when we substitute the 'x' and 'y' values from the ordered pair into the equation. It's like plugging in the suspects' information into our crime scene details and seeing if everything lines up. If it does, then the ordered pair is a solution! If it doesn't, well, then they're not our culprits.

The concept of ordered pairs and equations is fundamental in algebra and coordinate geometry. They allow us to represent relationships between two variables visually and algebraically. For example, the equation -6x + 7y = -6 represents a straight line when graphed on a coordinate plane. Each point on that line corresponds to an ordered pair that satisfies the equation. Understanding how to work with ordered pairs and equations is crucial for solving various mathematical problems, including finding intercepts, determining slopes, and solving systems of equations. Moreover, these concepts have practical applications in various fields, such as physics, engineering, and economics, where relationships between variables need to be analyzed and modeled. So, mastering this topic will definitely set you up for success in your mathematical journey!

Testing the Ordered Pair (1, 0)

Okay, let's put our first suspect, the ordered pair (1, 0), to the test. Remember, the first number in the ordered pair is our 'x' value, and the second number is our 'y' value. So, in this case, x = 1 and y = 0. Our mission is to substitute these values into our equation, -6x + 7y = -6, and see if the equation remains balanced. It's like we're weighing the evidence on a scale – both sides need to be equal for the ordered pair to be a solution. Let's plug in the values: -6(1) + 7(0) = -6. Now, let's simplify. -6 multiplied by 1 is -6, and 7 multiplied by 0 is 0. So, we have -6 + 0 = -6. This simplifies further to -6 = -6. Boom! The equation holds true! Both sides are equal. This means that when we substitute x = 1 and y = 0 into the equation, the equation is satisfied. So, the ordered pair (1, 0) is a solution to the equation -6x + 7y = -6. We've solved our first mystery!

The process of substituting values into an equation to check for solutions is a core technique in algebra. It's not just about plugging in numbers; it's about understanding the relationship between variables and how they interact within an equation. This skill is particularly useful when solving linear equations, inequalities, and systems of equations. By substituting potential solutions, we can quickly verify whether they satisfy the given conditions. Moreover, this method helps in identifying errors in our calculations and ensuring the accuracy of our results. Imagine you're building a bridge – you need to make sure that all the components fit together perfectly, right? Similarly, in mathematics, substituting values helps us ensure that our solutions fit the equation perfectly. This technique provides a solid foundation for tackling more complex mathematical problems in the future.

Testing the Ordered Pair (3, 12/7)

Alright, let's move on to our second suspect: the ordered pair (3, 12/7). This one looks a little more interesting because of the fraction, but don't worry, we've got this! Just like before, we need to substitute the 'x' and 'y' values into our equation, -6x + 7y = -6, and see if it holds true. This time, x = 3 and y = 12/7. Let's plug those values in: -6(3) + 7(12/7) = -6. Now, let's simplify. -6 multiplied by 3 is -18. For the second term, we have 7 multiplied by 12/7. Notice that the 7 in the numerator and the 7 in the denominator cancel each other out, leaving us with just 12. So, our equation now looks like this: -18 + 12 = -6. Let's simplify further. -18 plus 12 is -6. So, we have -6 = -6. Another match! The equation holds true. When we substitute x = 3 and y = 12/7 into the equation, the equation is satisfied. This means that the ordered pair (3, 12/7) is also a solution to the equation -6x + 7y = -6. We're on a roll, guys!

Dealing with fractions in equations might seem intimidating at first, but it's a crucial skill in mathematics. Fractions often appear in real-world problems, so mastering how to manipulate them within equations is essential. The key is to remember the rules of fraction multiplication and simplification. In this case, recognizing that the 7 in the numerator and denominator canceled out made the calculation much simpler. This highlights the importance of looking for opportunities to simplify expressions before diving into complex calculations. Furthermore, working with fractions enhances our understanding of number properties and strengthens our problem-solving abilities. Just think of fractions as puzzle pieces – once you understand how they fit together, you can solve any equation, no matter how fractional it may seem! This skill will not only help you in algebra but also in other areas of math and science where fractions are commonly used.

Conclusion: Case Closed!

So, after our investigation, what's the verdict? We've tested both ordered pairs, (1, 0) and (3, 12/7), and guess what? Both of them are solutions to the equation -6x + 7y = -6! We successfully substituted the x and y values from each ordered pair into the equation, simplified the expressions, and found that both equations balanced perfectly. It's like finding the missing pieces of a puzzle and seeing everything fit into place. This means that if we were to graph the equation -6x + 7y = -6, both of these points would lie on the line. Congratulations, we've solved our mathematical mystery for today! You've successfully determined whether ordered pairs are solutions to a linear equation. Great job, everyone!

This exercise demonstrates a fundamental concept in algebra: how to verify solutions to equations. It's a skill that you'll use repeatedly in various mathematical contexts. Understanding how to substitute values and check for equality is crucial for solving equations, graphing functions, and analyzing relationships between variables. Remember, mathematics is not just about finding the answer; it's about understanding the process and verifying your results. By mastering this skill, you're building a strong foundation for more advanced mathematical concepts. So, keep practicing, keep exploring, and keep those mathematical detective skills sharp! You've got the tools now, so go out there and solve some more equations!