To find the equation of line \( n \) that passes through the point \((-4, -1.5)\) and is parallel to the line that passes through the points \((0, 2)\) and \((5, 4)\), we first need to find the slope of the line formed by the points \((0, 2)\) and \((5, 4)\).
We can calculate the slope \( m \) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting in our points:
\[ m = \frac{4 - 2}{5 - 0} = \frac{2}{5} \]
Since line \( n \) is parallel to this line, it will have the same slope of \( \frac{2}{5} \).
Next, we use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
Where \((x_1, y_1)\) is the point through which the line passes. Substituting in the slope \( m = \frac{2}{5} \) and the point \((-4, -1.5)\):
\[ y - (-1.5) = \frac{2}{5}(x - (-4)) \]
This simplifies to:
\[ y + 1.5 = \frac{2}{5}(x + 4) \]
Now, distribute \( \frac{2}{5} \):
\[ y + 1.5 = \frac{2}{5}x + \frac{8}{5} \]
Next, we isolate \( y \) by subtracting \( 1.5 \) from both sides. Note that \( 1.5 = \frac{15}{10} \) can also be converted to a fraction for easy handling:
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Convert \( 1.5 \) to a fraction: \[ 1.5 = \frac{15}{10} \]
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Find a common denominator (the current form on the right can be simplified first):
To subtract \( 1.5 \) from \( \frac{8}{5} \):
\[ \frac{15}{10} = \frac{30}{20} \] Convert \( \frac{8}{5}\): \[ \frac{8}{5} = \frac{32}{20} \]
Finally: \[ y = \frac{2}{5}x + \frac{32}{20} - \frac{30}{20} = \frac{2}{5}x + \frac{2}{20} \] \[ y = \frac{2}{5}x + \frac{1}{10} \]
Putting it in a standard \( y = mx + b \) form:
The equation of the line \( n \) in slope-intercept form is:
\[ y = \frac{2}{5}x + \frac{1}{10} \]
If we need to rewrite this in the \( ? ?y = ? \) format, it can also be presented as:
\[ 2x - 5y + 1 = 0 \]
Therefore, enter
Complete the equation:
\[ 2x-5y= -1 \]
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