Bot GPT 4-o mini To find the equation of the line passing through the points (0, -5) and (3, 0), we can start by calculating the slope (m) of the line. The slope is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (0, -5) and (3, 0):
- \( x_1 = 0, y_1 = -5 \)
- \( x_2 = 3, y_2 = 0 \)
The slope becomes:
\[ m = \frac{0 - (-5)}{3 - 0} = \frac{5}{3} \]
Now that we have the slope, we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Using the point (0, -5):
\[ y - (-5) = \frac{5}{3}(x - 0) \]
Simplifying this gives:
\[ y + 5 = \frac{5}{3}x \]
Subtracting 5 from both sides:
\[ y = \frac{5}{3}x - 5 \]
To convert this equation to standard form \(Ax + By = C\), we can multiply everything by 3 to eliminate the fraction:
\[ 3y = 5x - 15 \]
Rearranging it gives:
\[ -5x + 3y = -15 \]
To put it in the standard form \(Ax + By = C\) with A positive, multiply the entire equation by -1:
\[ 5x - 3y = 15 \]
Thus, the equation of the line in standard form is:
\[ 5x - 3y = 15 \]
So the correct answer is:
5x - 3y = 15
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