The points $$ \mathrm{A}(5,4) $$ and $$ \mathrm{B}(9,-2) $$ lie on the blue line, labelled F , shown in the diagram above. Find an equation of the line $$ F $$.

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UpStudy AI · Accepted Answer

To find the equation of the line $$ F $$ that passes through the points $$ \mathrm{A}(5, 4) $$ and $$ \mathrm{B}(9, -2) $$, follow these steps:

1. **Calculate the Slope ($$ m $$)**:

The slope of a line passing through two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is given by:
   $$
   m = \frac{y_2 - y_1}{x_2 - x_1}
   $$
   
   Plugging in the coordinates of points $$ A $$ and $$ B $$:
   $$
   m = \frac{-2 - 4}{9 - 5} = \frac{-6}{4} = -\frac{3}{2}
   $$

2. **Find the Y-Intercept ($$ b $$)**:

Using the slope-intercept form $$ y = mx + b $$ and substituting one of the points (let's use $$ \mathrm{A}(5, 4) $$):
   $$
   4 = -\frac{3}{2} \cdot 5 + b
   $$
   $$
   4 = -\frac{15}{2} + b
   $$
   $$
   b = 4 + \frac{15}{2} = \frac{8}{2} + \frac{15}{2} = \frac{23}{2}
   $$

3. **Write the Equation of the Line**:

Combining the slope and y-intercept:
   $$
   y = -\frac{3}{2}x + \frac{23}{2}
   $$

**Final Answer:**
$$
y = -\frac{3}{2}\,x\ +\ \frac{23}{2}
$$
The equation of line $$ F $$ is $$ y = -\frac{3}{2}x + \frac{23}{2} $$.

The points and lie on the blue line,
labelled F , shown in the diagram above.
Find an equation of the line .

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The points and lie on the blue line, labelled F , shown in the diagram above. Find an equation of the line .