(1)) A line with a slope of $$ \frac{1}{9} $$ passes through the points $$ (r,-4) $$ and $$ (-2,-5) $$. What is th value of $$ r $$ ? $$ r=\square $$

Question

UpStudy AI · Accepted Answer

To find the value of $$ r $$, we can use the formula for the slope of a line, which is given by:

$$
m = \frac{y_2 - y_1}{x_2 - x_1}
$$

where $$ m $$ is the slope, and $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ are two points on the line.

In this case, we have the slope $$ m = \frac{1}{9} $$, the point $$ (r, -4) $$ as $$ (x_1, y_1) $$, and the point $$ (-2, -5) $$ as $$ (x_2, y_2) $$.

Substituting these values into the slope formula, we get:

$$
\frac{1}{9} = \frac{-5 - (-4)}{-2 - r}
$$

Now, let's simplify the equation step by step.

1. Simplify the numerator:
   $$
   -5 - (-4) = -5 + 4 = -1
   $$

2. Substitute this back into the equation:
   $$
   \frac{1}{9} = \frac{-1}{-2 - r}
   $$

3. This simplifies to:
   $$
   \frac{1}{9} = \frac{1}{2 + r}
   $$

4. Now, we can cross-multiply to solve for $$ r $$:
   $$
   1 \cdot (2 + r) = 9 \cdot 1
   $$

5. This gives us:
   $$
   2 + r = 9
   $$

6. Finally, solve for $$ r $$:
   $$
   r = 9 - 2 = 7
   $$

Thus, the value of $$ r $$ is:

$$
r = 7
$$
$$ r = 7 $$

(1)) A line with a slope of \( \frac{1}{9} \) passes through the points \( (r,-4) \) and \( (-2,-5) \). What is th value of \( r \) ? \( r=\square \)