Solving Linear Equations: Find (x, Y) For X - 3y = 9

Alright, let's dive into solving a simple linear equation! Our mission is to find an ordered pair (x, y) that makes the equation x - 3y = 9 true. This is a classic algebra problem, and we'll break it down step-by-step.

Understanding the Equation

First, let's understand what the equation x - 3y = 9 represents. This is a linear equation in two variables, x and y. A linear equation basically means that if you were to graph all the possible solutions (x, y) on a coordinate plane, you'd get a straight line. Each point on that line represents an ordered pair that satisfies the equation.

Our goal is to find just one of those points. Since there are infinitely many solutions (because there are infinitely many points on a line), we only need to find one pair of values for x and y that make the equation true.

Method 1: Choosing a Value for y and Solving for x

One of the easiest ways to find a solution is to pick a value for either x or y and then solve for the other variable. Let's start by choosing a value for y. A simple choice is often the best, so let's set y = 0. This simplifies our equation nicely.

Substituting y = 0 into the equation x - 3y = 9, we get:

x - 3(0) = 9

This simplifies to:

x - 0 = 9

Which further simplifies to:

x = 9

So, when y = 0, we find that x = 9. Therefore, the ordered pair (9, 0) is a solution to the equation x - 3y = 9. You can easily check this by plugging these values back into the original equation: 9 - 3(0) = 9, which is true.

Method 2: Choosing a Value for x and Solving for y

Alternatively, we could choose a value for x and solve for y. Let's try setting x = 0. Substituting x = 0 into the equation x - 3y = 9, we get:

0 - 3y = 9

This simplifies to:

-3y = 9

To solve for y, we divide both sides of the equation by -3:

y = 9 / -3

y = -3

So, when x = 0, we find that y = -3. Therefore, the ordered pair (0, -3) is another solution to the equation x - 3y = 9. Let's check this solution by plugging these values back into the original equation: 0 - 3(-3) = 0 + 9 = 9, which is also true.

Method 3: Rearranging the Equation

Another method involves rearranging the equation to solve for one variable in terms of the other. Let's solve for x in terms of y. Starting with x - 3y = 9, we add 3y to both sides of the equation:

x = 3y + 9

Now we can pick any value for y and easily find the corresponding value for x. For example, let's choose y = 1. Then:

x = 3(1) + 9

x = 3 + 9

x = 12

So, when y = 1, we find that x = 12. Therefore, the ordered pair (12, 1) is yet another solution to the equation x - 3y = 9. Checking this: 12 - 3(1) = 12 - 3 = 9, which is true.

We could also solve for y in terms of x. Starting with x - 3y = 9, we subtract x from both sides:

-3y = 9 - x

Then, we divide both sides by -3:

y = (9 - x) / -3

y = (x - 9) / 3

Now, we can choose any value for x and easily find the corresponding value for y. For example, let's choose x = 3. Then:

y = (3 - 9) / 3

y = -6 / 3

y = -2

So, when x = 3, we find that y = -2. Therefore, the ordered pair (3, -2) is a solution to the equation x - 3y = 9. Checking this: 3 - 3(-2) = 3 + 6 = 9, which is true.

Verifying the Solutions

It's always a good idea to verify your solutions by plugging them back into the original equation. This helps ensure that you haven't made any mistakes along the way.

• For the solution (9, 0): 9 - 3(0) = 9 - 0 = 9. This is correct.
• For the solution (0, -3): 0 - 3(-3) = 0 + 9 = 9. This is correct.
• For the solution (12, 1): 12 - 3(1) = 12 - 3 = 9. This is correct.
• For the solution (3, -2): 3 - 3(-2) = 3 + 6 = 9. This is correct.

Why Infinitely Many Solutions?

The equation x - 3y = 9 has infinitely many solutions because for any value you choose for x, you can find a corresponding value for y that satisfies the equation, and vice versa. Geometrically, this corresponds to the fact that a line extends infinitely in both directions. Each point on the line represents a solution to the equation.

Practical Applications

Understanding how to solve linear equations is a fundamental skill in algebra. It's used in various applications, such as:

• Modeling real-world situations: Linear equations can be used to model relationships between two variables, such as the relationship between the number of hours worked and the amount of money earned.
• Solving systems of equations: Many problems involve multiple equations with multiple variables. Solving systems of linear equations is a crucial skill in these scenarios.
• Graphing: Linear equations can be graphed on a coordinate plane, providing a visual representation of the relationship between the variables.

Conclusion

We've successfully found several ordered pairs (x, y) that are solutions to the equation x - 3y = 9. Remember, there are infinitely many solutions, and we've demonstrated a few different methods to find them. Whether you choose a value for x and solve for y, or vice versa, the key is to substitute the chosen value into the equation and solve for the remaining variable. Always verify your solution to ensure accuracy. Keep practicing, and you'll become a pro at solving linear equations in no time! Finding solutions to these equations is like unlocking doors in the world of mathematics – each solution opens up new possibilities and understanding.

So, to recap, here are the solutions we found:

• (9, 0)
• (0, -3)
• (12, 1)
• (3, -2)

Each of these ordered pairs satisfies the equation x - 3y = 9, and you can find countless others using the methods we've discussed. Happy solving!