Ordered Pairs: Does It Fit Y = -2/3 X + 6?

Hey math whizzes and number crunchers! Today, we're diving into the awesome world of relations and ordered pairs. Specifically, we're going to tackle a question that might seem a bit tricky at first glance: Which of the following is an ordered pair of the relation y=-rac{2}{3} x+6? We've got four options: A. $(3,4)$, B. ig(0,-rac{2}{3}ig), C. $(1,4)$, and D. $(3,8)$. Don't sweat it, guys! We're going to break this down step-by-step, making sure you totally get how to find the right fit for this equation. By the end of this, you'll be an ordered pair pro, ready to tackle any similar problem thrown your way. Let's get this math party started!

Understanding Relations and Ordered Pairs

Alright, let's get our heads around what we're dealing with here. A relation in mathematics is basically a set of ordered pairs. Think of it like a rule that connects inputs (usually the 'x' values) to outputs (usually the 'y' values). An ordered pair is just a pair of numbers written in a specific order, like $(x, y)$. The order is super important because the first number is always the input (x-coordinate) and the second number is always the output (y-coordinate). When we say an ordered pair belongs to a relation, it means that if you plug the 'x' value into the relation's equation, you'll get the corresponding 'y' value back. It's like a key and lock system – the ordered pair has to perfectly fit the equation's rule. Our specific relation here is y=-rac{2}{3} x+6. This equation tells us exactly how 'x' and 'y' are connected. For any 'x' we choose, we can calculate the 'y' that goes with it using this formula. Our job is to find which of the given ordered pairs $(x, y)$ satisfies this equation, meaning $y$ actually equals -rac{2}{3} x+6 when we plug in those specific numbers.

The Testing Ground: Plugging in the Values

So, how do we figure out which ordered pair is the true match for our equation, y=-rac{2}{3} x+6? It's actually pretty straightforward, guys! We just need to take each ordered pair and test it. For each option, we'll substitute the 'x' value from the pair into the equation and see if the calculated 'y' value matches the 'y' value given in the pair. If they match, bingo, we've found our answer! If they don't match, we move on to the next option. It's a process of elimination, but a fun one!

Let's start with option A: $(3,4)$. Here, $x=3$ and $y=4$. We plug $x=3$ into our equation: y = -rac{2}{3}(3) + 6. First, multiply -rac{2}{3} by $3$. The $3$s cancel out, leaving us with $-2$. So, the equation becomes $y = -2 + 6$. Adding $-2$ and $6$ gives us $4$. So, when $x=3$, our equation gives us $y=4$. Look at the ordered pair $(3,4)$ – the 'y' value is indeed $4$. Success! Option A works! However, in multiple-choice questions, it's always good practice to check the other options just to be absolutely sure, or if you made a calculation error. Plus, understanding why the others don't work can be super helpful for learning.

Now, let's check option B: ig(0,-rac{2}{3}ig). Here, $x=0$ and y=-rac{2}{3}. Let's plug $x=0$ into the equation: y = -rac{2}{3}(0) + 6. Anything multiplied by $0$ is $0$, so this simplifies to $y = 0 + 6$. This means $y=6$. The ordered pair is ig(0,-rac{2}{3}ig), but our calculation gave us $y=6$. Since -rac{2}{3} does not equal $6$, option B is not a match.

Moving on to option C: $(1,4)$. Here, $x=1$ and $y=4$. Let's substitute $x=1$ into the equation: y = -rac{2}{3}(1) + 6. Multiplying -rac{2}{3} by $1$ just gives us -rac{2}{3}. So, the equation becomes y = -rac{2}{3} + 6. To add these, we need a common denominator. We can write $6$ as rac{18}{3}. So, y = -rac{2}{3} + rac{18}{3}. Combining the numerators, we get y = rac{-2+18}{3}, which equals rac{16}{3}. The ordered pair is $(1,4)$, but our calculation resulted in y=rac{16}{3}. Since $4$ (or rac{12}{3}) does not equal rac{16}{3}, option C is not correct.

Finally, let's look at option D: $(3,8)$. Here, $x=3$ and $y=8$. We already tested $x=3$ when we looked at option A. Remember what we found? When $x=3$, the equation y = -rac{2}{3} x+6 gives us y = -rac{2}{3}(3) + 6 = -2 + 6 = 4. So, for $x=3$, the correct $y$ value in this relation is $4$, not $8$. Therefore, option D is also not a match.

The Verdict: A Clear Winner!

After going through all the options, what did we find, guys? We tested each ordered pair by plugging its 'x' value into the equation y=-rac{2}{3} x+6 and comparing the result to the 'y' value in the pair.

• For option A, $(3,4)$: When $x=3$, we calculated $y=4$. This matches the ordered pair!
• For option B, ig(0,-rac{2}{3}ig): When $x=0$, we calculated $y=6$. This does not match -rac{2}{3}.
• For option C, $(1,4)$: When $x=1$, we calculated y=rac{16}{3}. This does not match $4$.
• For option D, $(3,8)$: When $x=3$, we calculated $y=4$. This does not match $8$.

Based on our tests, only one ordered pair makes the equation true. That means option A. $(3,4)$ is the correct ordered pair that belongs to the relation y=-rac{2}{3} x+6. It's all about plugging in those numbers and seeing if the equation holds true! Keep practicing, and you'll master this in no time. High five!

Why Other Options Don't Work: A Deeper Dive

Let's take a moment to really understand why options B, C, and D are incorrect. This isn't just about getting the right answer; it's about solidifying your understanding of how equations and ordered pairs interact. When an ordered pair $(x, y)$ is said to be part of a relation defined by an equation, it means that substituting the specific $x$ and $y$ values into the equation will result in a true statement. If substituting the values leads to a false statement, then that ordered pair is not part of the relation. This concept is fundamental in graphing, solving systems of equations, and so much more in the world of mathematics.

For option B, ig(0,-rac{2}{3}ig), we had $x=0$. Plugging this into y=-rac{2}{3}x+6 gave us y = -rac{2}{3}(0) + 6 = 0 + 6 = 6. The equation says that when the input is $0$, the output must be $6$. The ordered pair proposes that the output is -rac{2}{3}. Since 6 eq -rac{2}{3}, this ordered pair doesn't satisfy the rule. You can think of the point $(0, 6)$ as the y-intercept of this line – it's where the graph crosses the y-axis. The point ig(0,-rac{2}{3}ig) would be on a different line.

Option C, $(1,4)$, had $x=1$. Our calculation yielded y = -rac{2}{3}(1) + 6 = -rac{2}{3} + rac{18}{3} = rac{16}{3}. The equation dictates that for an input of $1$, the output should be rac{16}{3}. The ordered pair suggests the output is $4$. Since 4 = rac{12}{3} and rac{12}{3} eq rac{16}{3}, this pair doesn't fit. This means the point (1, rac{16}{3}) would lie on the line represented by y=-rac{2}{3}x+6, but the point $(1,4)$ does not. It's slightly off!

Lastly, option D, $(3,8)$, shared the same x-value as our correct answer, $x=3$. When we tested $x=3$ for the relation y=-rac{2}{3}x+6, we found that the only y-value that makes the equation true is $y=4$. The ordered pair $(3,8)$ suggests that when $x=3$, $y$ should be $8$. This is a direct contradiction to what the equation requires. So, while $(3,4)$ is on the line, $(3,8)$ is not. It's important to remember that for a function (and this relation is a function), each input $x$ can only have one output $y$. The equation y=-rac{2}{3}x+6 defines that unique output for every $x$. Options B, C, and D fail this fundamental test of membership in the relation.

Visualizing the Relation: A Quick Sketch

To really nail this down, let's think about what this relation y=-rac{2}{3}x+6 looks like graphically. This is the equation of a straight line. The number $6$ is the y-intercept (where the line crosses the y-axis), so we know the line passes through the point $(0,6)$. The coefficient -rac{2}{3} is the slope. This means for every $3$ units you move to the right (the 'run'), you move down $2$ units (the 'rise').

Let's plot our correct point, $(3,4)$. Starting from the origin $(0,0)$, you'd go right $3$ units and up $4$ units. Does this point lie on the line? Let's use the slope. From the y-intercept $(0,6)$, if we move $3$ units right (to $x=3$), we should move $2$ units down (to $y = 6-2 = 4$). So, the point $(3,4)$ is indeed on the line! This confirms our algebraic solution.

Now, consider the other points:

• (0, -rac{2}{3}): This point is on the y-axis, but it's below the origin. Our line crosses the y-axis at $(0,6)$, so this point is nowhere near our line.
• $(1,4)$: This point is to the right of the y-axis and up. If we found y=rac{16}{3} for $x=1$, that's approximately $5.33$. So, the point $(1, 5.33)$ is on the line. The point $(1,4)$ is below where the line would be.
• $(3,8)$: We know $(3,4)$ is on the line. Since $8$ is greater than $4$, the point $(3,8)$ is above our line at the x-coordinate of $3$.

Visualizing it like this helps you see that only one point can truly satisfy the equation for a given $x$ value in a linear relation. It's all about staying on that specific line!

Conclusion: Mastering Ordered Pairs

So there you have it, math enthusiasts! We've systematically checked each ordered pair against the relation y=-rac{2}{3} x+6. By substituting the x-values and calculating the corresponding y-values, we were able to determine which pair satisfied the equation. Remember, the key is to plug and play! Test each option, and if the equation holds true, you've found your match. In this case, option A. $(3,4)$ is the only ordered pair that fits perfectly into the relation y=-rac{2}{3} x+6. Keep practicing these types of problems, and you'll become a pro at identifying ordered pairs that belong to any given relation. It's a fundamental skill that will serve you well as you continue your mathematical journey. Keep up the great work, and happy calculating!