To find the equation of the line that passes through the points (3, -2) and (6, 2), we can follow these steps:
a. Write an equation for the line in point-slope form.
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Calculate the slope (m) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the points (3, -2) and (6, 2): \[ m = \frac{2 - (-2)}{6 - 3} = \frac{2 + 2}{3} = \frac{4}{3} \]
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Use the point-slope form of a line, which is: \[ y - y_1 = m(x - x_1) \] Using one of the points, say (3, -2): \[ y - (-2) = \frac{4}{3}(x - 3) \] This simplifies to: \[ y + 2 = \frac{4}{3}(x - 3) \]
Thus, the equation in point-slope form is: \[ y + 2 = \frac{4}{3}(x - 3) \]
b. Rewrite the equation in standard form using integers.
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Distribute the slope on the right side: \[ y + 2 = \frac{4}{3}x - 4 \]
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Clear the fraction by multiplying the entire equation by 3 (the denominator): \[ 3(y + 2) = 4x - 12 \] This gives: \[ 3y + 6 = 4x - 12 \]
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Rearranging to standard form \(Ax + By = C\): \[ -4x + 3y + 6 + 12 = 0\ \Rightarrow\ -4x + 3y + 18 = 0 \] To put it in the standard form: \[ 4x - 3y = 18 \]
Thus, the standard form of the equation is: \[ 4x - 3y = 18 \]
Summary:
- Point-Slope Form: \( y + 2 = \frac{4}{3}(x - 3) \)
- Standard Form: \( 4x - 3y = 18 \)
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