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"Los puntos \((-2, r)\) y \(4, -10)\) se encuentran en una recta con pendiente \(m = 2\). Hallar la coordenada desconocida \(r\)."

We are given two points, \((-2, r)\) and \( (4, -10)\), which lie on a line with slope \(m = 2\), and we need to find the unknown coordinate \(r\).

The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the values:

\[
2 = \frac{-10 - r}{4 - (-2)}
\]

Simplifying the denominator:

\[
2 = \frac{-10 - r}{6}
\]

Now, multiply both sides by 6:

\[
12 = -10 - r
\]

Solve for \(r\):

\[
r = -10 - 12 = -22
\]

Thus, the unknown coordinate \(r\) is \(-22\).

Let me know if you would like more details or have any questions!

Here are 5 related questions to expand your understanding:

1. How would the solution change if the slope was negative?
2. What is the general equation of a line in slope-intercept form?
3. How do you find the midpoint between two points on a line?
4. What is the equation of the line passing through the two given points?
5. How can we use the distance formula to find the length of a line segment between two points?

**Tip**: Understanding the slope formula is key in geometry problems involving lines and helps in finding missing coordinates easily.

Los puntos (-2, r) y (4, -10) se encuentran en una recta con pendiente m = 2. Hallar la coordenada desconocida r.

Learn how to find the unknown coordinate r in the points (-2, r) and (4, -10) on a line with slope m = 2. This problem covers key algebraic concepts, including the slope formula and solving for unknowns. Suitable for students in Grades 8-10.

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