Question

Vladimir says that the equation of the line that passes through points (negative 5, negative 3) and (10, 9) is y = four-fifths x + 1. Robyn says that the line passes through the points (negative 10, negative 7) and (negative 15, negative 11). Who is correct?
only Vladimir
only Robyn
both Vladimir and Robyn
neither Vladimir nor Robyn

Answer 1

To determine who is correct regarding the equations of the line and the points through which it passes, we can check both Vladimir's and Robyn's claims.

**Step 1: Calculate the slope and y-intercept of the line through the points (-5, -3) and (10, 9)**.

**Slope Calculation**:
The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points (-5, -3) and (10, 9):
\[
m = \frac{9 - (-3)}{10 - (-5)} = \frac{9 + 3}{10 + 5} = \frac{12}{15} = \frac{4}{5}
\]

**Y-Intercept Calculation**:
Now that we have the slope \(m = \frac{4}{5}\), we can use point-slope form to find the y-intercept (b). The equation of a line in slope-intercept form is \(y = mx + b\).

Using point (-5, -3):
\[
-3 = \frac{4}{5}(-5) + b
\]
\[
-3 = -4 + b
\]
\[
b = -3 + 4 = 1
\]

Thus, the equation of the line that passes through the points (-5, -3) and (10, 9) is:
\[
y = \frac{4}{5}x + 1
\]

**Conclusion about Vladimir's Claim**:
Vladimir is correct; the equation \(y = \frac{4}{5}x + 1\) is verified.

**Step 2: Check Robyn's claim that the line passes through points (-10, -7) and (-15, -11)**.

Calculate the slope between (-10, -7) and (-15, -11):
\[
m = \frac{-11 - (-7)}{-15 - (-10)} = \frac{-11 + 7}{-15 + 10} = \frac{-4}{-5} = \frac{4}{5}
\]

The slope is \(\frac{4}{5}\) which matches Vladimir's slope. Now we check if it gives the same y-intercept using, for example, point (-10, -7):
\[
-7 = \frac{4}{5}(-10) + b
\]
\[
-7 = -8 + b
\]
\[
b = -7 + 8 = 1
\]

This gives the same y-intercept of 1, confirming the line given by both sets of points is:
\[
y = \frac{4}{5}x + 1
\]

**Conclusion about Robyn's Claim**:
Robyn's claim is also correct; both sets of points lie on the same line.

Since both Vladimir and Robyn are correct:

The answer is **both Vladimir and Robyn**.