To find the equation of the line passing through the points \((-2.7, 0.78)\) and \((21, 15)\), we will follow these steps:
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Calculate the slope (m): The slope \( m \) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( (x_1, y_1) = (-2.7, 0.78) \) and \( (x_2, y_2) = (21, 15) \).
Substituting the values, we have: \[ m = \frac{15 - 0.78}{21 - (-2.7)} = \frac{15 - 0.78}{21 + 2.7} = \frac{14.22}{23.7} \]
Now we calculate \( \frac{14.22}{23.7} \): \[ m \approx 0.599 \text{ (approximating to three decimal places)} \]
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Use the point-slope form of the equation of a line: The point-slope form is given by: \[ y - y_1 = m(x - x_1) \]
We can use point \((-2.7, 0.78)\): \[ y - 0.78 = 0.599(x + 2.7) \]
Now expanding this equation: \[ y - 0.78 = 0.599x + 0.599 \cdot 2.7 \] Calculate \( 0.599 \cdot 2.7 \): \[ 0.599 \cdot 2.7 \approx 1.6133 \] Thus, we have: \[ y - 0.78 = 0.599x + 1.6133 \] Adding \( 0.78 \) to both sides: \[ y = 0.599x + 1.6133 + 0.78 \] \[ y = 0.599x + 2.3933 \]
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Convert to standard form \( Ax + By = C \): We start from the slope-intercept form: \[ y = 0.599x + 2.3933 \] Rearranging gives: \[ -0.599x + y = 2.3933 \] To convert to standard form (preferably with integer coefficients), we multiply through by 1000 to eliminate the decimal: \[ -599x + 1000y = 2393.3 \]
We can multiply everything by 10 to make it easier: \[ -5990x + 10000y = 23933 \]
To make the leading coefficient positive \( A = 5990 \): \[ 5990x - 10000y = -23933 \]
Therefore, the equation of the line in standard form is: \[ 5990x - 10000y = -23933 \]
To summarize, the final answer in standard form is: \[ 5990x - 10000y = -23933 \]