Questions: (A) Find the slope of the line that passes through the given points. (B) Find the point-slope form of the equation of the line. (C) Find the slope-intercept form of the equation of the line. (D) Find the standard form of the equation of the line. (4,1) and (7,3)

Solution Steps

To find the slope \( m \) of the line passing through the points \( (4, 1) \) and \( (7, 3) \), use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the given points: \[ m = \frac{3 - 1}{7 - 4} = \frac{2}{3} \] The slope is \( \boxed{m = \frac{2}{3}} \).

The point-slope form of a line is: \[ y - y_1 = m(x - x_1) \] Using the slope \( m = \frac{2}{3} \) and the point \( (4, 1) \): \[ y - 1 = \frac{2}{3}(x - 4) \] The point-slope form is \( \boxed{y - 1 = \frac{2}{3}(x - 4)} \).

The slope-intercept form of a line is: \[ y = mx + b \] Start with the point-slope form and solve for \( y \): \[ y - 1 = \frac{2}{3}(x - 4) \] \[ y = \frac{2}{3}x - \frac{8}{3} + 1 \] \[ y = \frac{2}{3}x - \frac{5}{3} \] The slope-intercept form is \( \boxed{y = \frac{2}{3}x - \frac{5}{3}} \).

Final Answer