To find the equation of the line in point-slope form, we first need to calculate the slope (\(m\)) of the line that passes through the points \((-7, -5)\) and \((-5, 4)\).
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of the points:
\[ m = \frac{4 - (-5)}{-5 - (-7)} = \frac{4 + 5}{-5 + 7} = \frac{9}{2} \]
Now that we have the slope, we can use one of the points (let's use the point \((-7, -5)\)) to write the equation in point-slope form. The point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
Substituting the slope and the coordinates of the point:
\[ y - (-5) = \frac{9}{2}(x - (-7)) \]
Simplifying this, we have:
\[ y + 5 = \frac{9}{2}(x + 7) \]
This is the equation of the line in point-slope form.
Next, we will rewrite the equation in standard form. First, we will distribute and simplify the point-slope equation:
\[ y + 5 = \frac{9}{2}(x + 7) \]
Distributing \(\frac{9}{2}\) on the right:
\[ y + 5 = \frac{9}{2}x + \frac{63}{2} \]
Now, we will subtract \(5\) from both sides. Since \(5\) can be written as \(\frac{10}{2}\), we can write the equation as:
\[ y = \frac{9}{2}x + \frac{63}{2} - \frac{10}{2} \]
This simplifies to:
\[ y = \frac{9}{2}x + \frac{53}{2} \]
Next, to convert this to standard form \(Ax + By = C\) using integers, we can multiply through by \(2\) to eliminate the fractions:
\[ 2y = 9x + 53 \]
Rearranging it into standard form, we get:
\[ -9x + 2y = 53 \]
To follow the convention of \(A\) being positive, we can multiply the entire equation by \(-1\):
\[ 9x - 2y = -53 \]
Thus, the final standard form of the equation is:
\[ 9x - 2y = -53 \]
So the required equations are:
- Point-slope form: \(y + 5 = \frac{9}{2}(x + 7)\)
- Standard form: \(9x - 2y = -53\)
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