Line Equation Showdown: Who's Right, Vladimir Or Robyn?

Nov 17, 2025 by Admin 56 views

Hey guys, let's dive into a classic math problem! We're talking about lines, equations, and who's got it right. Vladimir and Robyn both have opinions on the equation of a line, and we need to figure out who's on the money. This is a great exercise in understanding how to find the equation of a line and verifying if a given point actually sits on that line. So, let's get to it and break down each person's claim, step by step, to see who nails the equation of the line. We'll use our math skills to see if their equations and points hold up.

Vladimir's Claim: The Equation Through Two Points

Vladimir says the line that goes through the points $(-5, -3)$ and $(10, 9)$ has the equation $y = rac{4}{5}x + 1$ . Alright, let's put Vladimir's claim to the test. To do this, we need to first confirm the equation is correct by using the two points he gave, then we verify that the points lie on the equation. The core concept here is using the slope-intercept form ( $y = mx + b$ ) to determine the equation of a line. We first must find the slope, 'm', and then the y-intercept, 'b'. Let's start with finding the slope. The slope formula is: $m = (y_2 - y_1) / (x_2 - x_1)$ .

Using Vladimir's points, $(-5, -3)$ as $(x_1, y_1)$ and $(10, 9)$ as $(x_2, y_2)$ , we get:

$m = (9 - (-3)) / (10 - (-5))$ $m = (9 + 3) / (10 + 5)$ $m = 12 / 15$ $m = 4 / 5$

So far, so good! The slope Vladimir claimed is correct. Now that we've found the slope, we have $y = rac{4}{5}x + b$ . To find 'b', the y-intercept, we'll plug in one of the points into the equation. Let's use $(10, 9)$ :

$9 = ( rac{4}{5} * 10) + b$ $9 = 8 + b$ $b = 1$

Perfect! The y-intercept 'b' is 1. Therefore, the equation of the line is indeed $y = rac{4}{5}x + 1$ . Now, we must check if Vladimir's point $(-5, -3)$ lies on the line:

$-3 = ( rac{4}{5} * -5) + 1$ $-3 = -4 + 1$ $-3 = -3

It is correct! So, Vladimir's equation and the points provided checks out. It looks like he's on the right track! But, we must check Robyn's claim.

Validating Vladimir's Equation and Points

To ensure Vladimir is absolutely correct, let's confirm the points he used actually lie on the line described by his equation. We have already found the equation to be $y = rac{4}{5}x + 1$ and we know that the first point is $(-5,-3)$ . To confirm the point lies on the line, we substitute the x and y values from the point into the equation to see if the equation holds true. Substituting x = -5 and y = -3:

$-3 = rac{4}{5}(-5) + 1$ $-3 = -4 + 1$ $-3 = -3

This confirms that the point (-5, -3) does indeed lie on the line. Now, we must do the same for the second point, which is (10, 9). Substituting x = 10 and y = 9:

$9 = rac{4}{5}(10) + 1$ $9 = 8 + 1$ $9 = 9$

This also confirms that the point (10, 9) lies on the line. Since Vladimir's equation aligns perfectly with the provided points, it's safe to say he's correct! Next, let's examine Robyn's assertion and see if it also holds up to scrutiny.

Robyn's Claim: Another Set of Points

Now, let's see what Robyn has to say. Robyn claims the same line passes through the points $(-10, -7)$ and $(-15, -11)$ . To check Robyn's claim, we can use the same equation we found from Vladimir's points, $y = rac{4}{5}x + 1$ . Because if Robyn is correct, then her points must also satisfy this equation. We'll plug in Robyn's points to see if they fit the bill. Let's start with the point $(-10, -7)$ . Substituting x = -10 and y = -7 into the equation:

$-7 = rac{4}{5}(-10) + 1$ $-7 = -8 + 1$ $-7 = -7

The first point checks out! It looks like Robyn might be on to something. Now, let's check the second point, $(-15, -11)$ . Substituting x = -15 and y = -11 into the equation:

$-11 = rac{4}{5}(-15) + 1$ $-11 = -12 + 1$ $-11 = -11

Both of Robyn's points satisfy the equation. This is where it gets interesting. Both Vladimir and Robyn seem to be correct! But wait, could there be a catch? Are they both talking about the same line?

Validating Robyn's Equation and Points

To be absolutely sure about Robyn's claim, we have to verify that her points $(-10, -7)$ and $(-15, -11)$ actually fall on the line we previously established. Remember, Vladimir's points gave us the equation $y = rac{4}{5}x + 1$ . The most straightforward approach is to test Robyn's points against this established equation.

Let's start with the point $(-10, -7)$ . Plugging these values into the equation gives us:

$-7 = rac{4}{5}(-10) + 1$ $-7 = -8 + 1$ $-7 = -7

This confirms that the point $(-10, -7)$ does indeed satisfy the equation. Next, let's validate the second point, $(-15, -11)$ . Substituting these values into the same equation:

$-11 = rac{4}{5}(-15) + 1$ $-11 = -12 + 1$ $-11 = -11

Again, the equation holds true! This means that the point $(-15, -11)$ also lies on the line defined by $y = rac{4}{5}x + 1$ . Since both of Robyn's points satisfy the same equation derived from Vladimir's points, it is confirmed that they are on the same line. In this scenario, both are technically correct because they provided points that satisfy the same linear equation. It is very important to see the relationship between points and equations.

The Verdict: Who is Correct?

So, who is correct? The answer is both! Both Vladimir and Robyn are correct. Vladimir gave us the correct equation and points that satisfied the equation, and Robyn provided points that also satisfied the same equation. All of the points given lie on the same line, which is represented by the equation $y = rac{4}{5}x + 1$ . They are just using different points on the same line to prove their points.

This problem highlights the importance of understanding the relationship between points, slopes, and linear equations. The equation of a line is unique, but it can be defined by different sets of points that all lie on that same line. This is a common situation you’ll find in math, where multiple approaches can lead to the same correct solution. Keep up the great work in your math journey, guys!