Solving Linear Equations: Find Ordered Pairs That Fit

Have you ever wondered how to find the points that perfectly fit on the line of a linear equation? It's like finding the right keys to unlock a door! In this article, we'll explore how to identify ordered pairs that are solutions to the equation -6x - 5y = 5. We'll break it down step by step, making it super easy to understand. So, let's dive in and become equation-solving pros!

Understanding Ordered Pairs and Linear Equations

Before we jump into solving, let's make sure we're on the same page about what ordered pairs and linear equations are. Think of ordered pairs as coordinates on a map – they tell you exactly where to find a specific spot. They're written as (x, y), where 'x' is the horizontal position and 'y' is the vertical position. Linear equations, on the other hand, are like the directions to draw a straight line on that map. They have the form Ax + By = C, where A, B, and C are constants.

What Makes an Ordered Pair a Solution?

Now, here's the key: an ordered pair is a solution to a linear equation if, when you plug in the x and y values into the equation, it makes the equation true. It's like finding the perfect puzzle piece that fits just right. If the equation balances out after you substitute the values, then that ordered pair is a point on the line represented by the equation. If it doesn't, then it's not a solution. This is the fundamental concept we'll use to solve our problem. We'll take each ordered pair, plug its x and y values into the equation -6x - 5y = 5, and see if the equation holds true. This process of substitution and evaluation is the heart of determining whether a point lies on the line.

Why This Matters

Understanding how to find solutions to linear equations isn't just a math exercise; it's a crucial skill with real-world applications. Linear equations are used to model countless situations, from calculating the cost of a service based on usage to predicting trends and making informed decisions. Being able to identify ordered pairs that satisfy an equation allows us to pinpoint specific points on a graph, which can represent anything from the break-even point in a business to the optimal combination of resources in a project. Moreover, the ability to manipulate and solve linear equations forms the foundation for more advanced mathematical concepts, such as systems of equations and linear programming, which are essential tools in fields like engineering, economics, and computer science. So, mastering this skill is not just about getting the right answer on a test; it's about developing a powerful problem-solving tool that will serve you well in various aspects of life and future studies.

Testing the Ordered Pairs

Alright, let's put our knowledge to the test! We have the equation -6x - 5y = 5, and we're going to check a few ordered pairs to see if they're solutions. We'll go through each pair one by one, showing you exactly how to plug in the x and y values and simplify. Get ready to see some math magic happen!

A. (0, 6)

Our first contender is the ordered pair (0, 6). Remember, the first number is 'x' and the second is 'y'. So, let's substitute x = 0 and y = 6 into our equation:

-6(0) - 5(6) = 5

Now, let's simplify. -6 times 0 is 0, and -5 times 6 is -30. So we have:

0 - 30 = 5

Which simplifies to:

-30 = 5

Hmm, -30 does not equal 5. This means the ordered pair (0, 6) is not a solution to the equation. It's like trying to fit the wrong puzzle piece – it just doesn't work!

B. (7, 5)

Next up, we have the ordered pair (7, 5). Let's plug in x = 7 and y = 5:

-6(7) - 5(5) = 5

Simplifying, -6 times 7 is -42, and -5 times 5 is -25. Our equation now looks like this:

-42 - 25 = 5

Which simplifies to:

-67 = 5

Again, -67 is definitely not equal to 5. So, (7, 5) is also not a solution. We're on a roll finding pairs that don't fit!

C. (-5, 5)

Let's keep going with the ordered pair (-5, 5). Substituting x = -5 and y = 5, we get:

-6(-5) - 5(5) = 5

Now, -6 times -5 is 30 (remember, a negative times a negative is a positive!), and -5 times 5 is -25. So our equation becomes:

30 - 25 = 5

Which simplifies to:

5 = 5

Hey, look at that! 5 does equal 5. This means the ordered pair (-5, 5) is a solution to the equation. We've found our first match!

D. (5, -7)

Moving on to (5, -7), let's plug in x = 5 and y = -7:

-6(5) - 5(-7) = 5

Simplifying, -6 times 5 is -30, and -5 times -7 is 35. Our equation now looks like this:

-30 + 35 = 5

Which simplifies to:

5 = 5

Another match! 5 equals 5, so the ordered pair (5, -7) is a solution.

E. (-7, 1)

Let's try (-7, 1). Substituting x = -7 and y = 1, we have:

-6(-7) - 5(1) = 5

Simplifying, -6 times -7 is 42, and -5 times 1 is -5. So our equation becomes:

42 - 5 = 5

Which simplifies to:

37 = 5

Nope, 37 does not equal 5. Therefore, (-7, 1) is not a solution.

F. (0, -1)

Last but not least, let's check (0, -1). Plugging in x = 0 and y = -1, we get:

-6(0) - 5(-1) = 5

Simplifying, -6 times 0 is 0, and -5 times -1 is 5. Our equation becomes:

0 + 5 = 5

Which simplifies to:

5 = 5

Yes! 5 equals 5, so the ordered pair (0, -1) is a solution.

Reviewing Our Findings

After carefully testing each ordered pair, we found that the following are solutions to the equation -6x - 5y = 5:

• (-5, 5)
• (5, -7)
• (0, -1)

The other pairs, (0, 6), (7, 5), and (-7, 1), did not satisfy the equation, so they are not points on the line.

Graphing the Equation and Solutions

To really understand what's going on, let's visualize the equation and its solutions on a graph. Imagine a coordinate plane with the x-axis running horizontally and the y-axis running vertically. Our equation, -6x - 5y = 5, represents a straight line on this plane. The ordered pairs that are solutions to the equation are the points that lie exactly on this line.

Plotting the Points

We found three solutions: (-5, 5), (5, -7), and (0, -1). Let's plot these points on our imaginary graph.

• (-5, 5): Start at the origin (0, 0), move 5 units to the left on the x-axis, and then 5 units up on the y-axis. Mark that point.
• (5, -7): Start at the origin, move 5 units to the right on the x-axis, and then 7 units down on the y-axis. Mark that point.
• (0, -1): Start at the origin, don't move on the x-axis (since x is 0), and move 1 unit down on the y-axis. Mark that point.

Drawing the Line

Now, if you were to take a ruler and draw a straight line through these three points, you would see the visual representation of the equation -6x - 5y = 5. This line extends infinitely in both directions, and every single point on this line is a solution to the equation. The points we didn't select, like (0, 6) and (7, 5), would be located off the line, demonstrating why they are not solutions.

The Power of Visualization

Graphing an equation and its solutions is a powerful way to understand the relationship between algebra and geometry. It allows you to see how an equation translates into a visual form, and how solutions are simply points that fit on that form. This visual understanding can make solving equations much more intuitive and can help you grasp more advanced mathematical concepts in the future.

Conclusion

And there you have it! We've successfully identified the ordered pairs that are solutions to the equation -6x - 5y = 5. We learned how to substitute values, simplify equations, and determine whether a pair fits the equation. Remember, practice makes perfect, so keep solving and exploring! You're on your way to becoming a math whiz! If you want to delve deeper into linear equations and graphing, check out resources like Khan Academy's Linear Equations Section for more examples and explanations.