To find the equation of the line that passes through the points \((-7, -5)\) and \((-5, 4)\), we will first calculate the slope of the line, and then formulate the equation in point-slope form and standard form.
a. Point-Slope Form
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Calculate the slope (m):
The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points \((-7, -5)\) and \((-5, 4)\): \[ m = \frac{4 - (-5)}{-5 - (-7)} = \frac{4 + 5}{-5 + 7} = \frac{9}{2} \]
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Use point-slope form:
The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] Using point \((-7, -5)\) and the slope \( m = \frac{9}{2} \): \[ y - (-5) = \frac{9}{2}(x - (-7)) \] Simplifying this: \[ y + 5 = \frac{9}{2}(x + 7) \] So, the equation in point-slope form is: \[ y + 5 = \frac{9}{2}(x + 7) \]
b. Standard Form
To convert the point-slope form equation into standard form, we can start with: \[ y + 5 = \frac{9}{2}(x + 7) \]
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Clear the fraction by multiplying through by 2: \[ 2(y + 5) = 9(x + 7) \] Which simplifies to: \[ 2y + 10 = 9x + 63 \]
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Rearrange into standard form \(Ax + By = C\):
Move all terms to one side: \[ -9x + 2y = 63 - 10 \] This simplifies to: \[ -9x + 2y = 53 \]
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Multiply through by -1 to make the leading coefficient positive (if required): \[ 9x - 2y = -53 \]
In standard form, we have: \[ 9x - 2y = -53 \]
Final Answers:
a. Point-slope form: \( y + 5 = \frac{9}{2}(x + 7) \)
b. Standard form: \( 9x - 2y = -53 \)