Graphing Equations: Find Points On The Line

Hey guys! Today, we're diving into the super cool world of coordinate geometry, specifically how to figure out if a point actually lives on the graph of an equation. You know, like checking if a friend's house is on a specific street. Our main mission is to solve the problem: Which ordered pairs represent points on the graph of this equation? Select all that apply. $2 x-5 y=11$ We've got a few suspects: A. $(3,-1)$, B. $(5,7)$, C. $(-8,3)$, D. $(8,1)$, E. $(-2,-3)$, and F. $(-7,-5)$. Let's put on our detective hats and check each one to see if it fits the bill. Finding points on the graph of an equation is all about substitution. You take the x and y values from the ordered pair and plug them into the equation. If the equation holds true (meaning the left side equals the right side), then BAM! That point is definitely on the graph. If not, well, it's hanging out somewhere else in the coordinate plane. It's a pretty straightforward process, and once you get the hang of it, you'll be spotting points on graphs like a pro. We're going to go through each option methodically, so stick around and let's solve this puzzle together!

Testing Point A: (3, -1)

Alright, first up is point A. $(3,-1)$. Remember, the equation we're working with is $2x - 5y = 11$. For this point, our 'x' value is 3, and our 'y' value is -1. Let's substitute these into the equation and see what happens. We'll replace 'x' with 3 and 'y' with -1. So, we get: $2(3) - 5(-1)$. Now, let's crunch these numbers. $2$ times $3$ is $6$. And $5$ times $-1$ is $-5$. So, the expression becomes $6 - (-5)$. Subtracting a negative is the same as adding a positive, right? So, $6 - (-5)$ is the same as $6 + 5$. And what does $6 + 5$ equal? You guessed it: $11$. Now, we compare this result to the right side of our original equation, which is also $11$. Since $11 = 11$, this statement is true. This means that the ordered pair $(3,-1)$ is a point on the graph of the equation $2x - 5y = 11$. So, we can definitely tick Option A off our list as a correct answer. High five! It’s always a win when the first try works out, but don't get too comfortable, we have more points to check, and you never know what surprises await us!

Testing Point B: (5, 7)

Moving on to our next suspect, B. $(5,7)$. Here, $x=5$ and $y=7$. We'll plug these values into our trusty equation: $2x - 5y = 11$. Substituting, we get $2(5) - 5(7)$. Let's do the math. $2$ times $5$ is $10$. And $5$ times $7$ is $35$. So, our expression is $10 - 35$. What is $10 - 35$? It's $-25$. Now, we compare this result to the right side of our equation, which is $11$. Is $-25$ equal to $11$? Nope! This statement is false. Therefore, the ordered pair $(5,7)$ is not a point on the graph of $2x - 5y = 11$. We can confidently say that Option B is not a correct answer. Keep those detective skills sharp, guys, because we're just getting started!

Testing Point C: (-8, 3)

Next on our investigation list is C. $(-8,3)$. For this point, our 'x' is $-8$ and our 'y' is $3$. Let's substitute these into the equation $2x - 5y = 11$. We get $2(-8) - 5(3)$. Time for some calculations. $2$ times $-8$ is $-16$. And $5$ times $3$ is $15$. So, the expression becomes $-16 - 15$. What is $-16 - 15$? It equals $-31$. Now, we compare $-31$ to our target number, $11$. Are they the same? Absolutely not. $-31 eq 11$. So, the ordered pair $(-8,3)$ does not lie on the graph of $2x - 5y = 11$. Option C is a bust. Don't let that get you down, though. The more points we check, the closer we get to the full picture. Keep your focus, and we'll nail this!

Testing Point D: (8, 1)

Let's check out D. $(8,1)$. Here, $x=8$ and $y=1$. Plugging these into $2x - 5y = 11$, we get $2(8) - 5(1)$. Let's multiply. $2$ times $8$ is $16$. And $5$ times $1$ is $5$. So, we have $16 - 5$. What's $16 - 5$? It's $11$. Now, we compare this $11$ to the right side of our equation, which is also $11$. Since $11 = 11$, this statement is true. This means that the ordered pair $(8,1)$ is a point on the graph of the equation $2x - 5y = 11$. So, we can add Option D to our list of correct answers. Nice work, everyone! We're on a roll now, finding more and more points that belong to our equation's graph.

Testing Point E: (-2, -3)

Alright, let's tackle E. $(-2,-3)$. For this point, $x = -2$ and $y = -3$. Substituting into $2x - 5y = 11$, we get $2(-2) - 5(-3)$. Let's calculate. $2$ times $-2$ is $-4$. And $5$ times $-3$ is $-15$. So, we have $-4 - (-15)$. Remember, subtracting a negative is like adding a positive. So, $-4 - (-15)$ becomes $-4 + 15$. What is $-4 + 15$? It's $11$. Now, we compare this $11$ to the right side of our equation, which is $11$. Since $11 = 11$, this statement is true. This confirms that the ordered pair $(-2,-3)$ is a point on the graph of $2x - 5y = 11$. Let's add Option E to our collection of correct answers. We're getting really close to solving this!

Testing Point F: (-7, -5)

Finally, let's examine F. $(-7,-5)$. In this case, $x = -7$ and $y = -5$. Let's substitute these into the equation $2x - 5y = 11$. We get $2(-7) - 5(-5)$. Time for the arithmetic. $2$ times $-7$ is $-14$. And $5$ times $-5$ is $-25$. So, our expression is $-14 - (-25)$. Again, subtracting a negative means adding a positive: $-14 + 25$. What is $-14 + 25$? It equals $11$. We compare this result to the right side of our equation, $11$. Since $11 = 11$, this statement is true. Therefore, the ordered pair $(-7,-5)$ is a point on the graph of $2x - 5y = 11$. We can proudly add Option F to our list of correct answers. Fantastic job, everyone!

Conclusion: Which Points Are On The Graph?

So, after all that detective work, guys, we've checked every single ordered pair against the equation $2x - 5y = 11$. We found that points A, D, E, and F made the equation true when we substituted their x and y values. This means these are the points that actually lie on the graph of the line represented by this equation. The points that do represent points on the graph of the equation $2x - 5y = 11$ are: A. $(3,-1)$, D. $(8,1)$, E. $(-2,-3)$, and F. $(-7,-5)$. Remember, the key to this whole process is substitution and checking for equality. Keep practicing these skills, and you'll master graphing and identifying points on lines in no time. Keep up the great work, mathematicians!