Which equation represents the line that passes through the points \( (4,1) \) and \( (0,3) \) ? (A) \( y+1=-\frac{1}{2}(x+4) \) B \( y-1=-\frac{1}{2}(x-4) \) C \( y-4=-\frac{1}{2}(x-1) \) (D) \( y+4=-\frac{1}{2}(x+1) \)

To find the equation of the line that passes through the points \( (4,1) \) and \( (0,3) \), we start by calculating the slope \( m \). The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where we can take \( (x_1, y_1) = (4, 1) \) and \( (x_2, y_2) = (0, 3) \). So, \( m = \frac{3 - 1}{0 - 4} = \frac{2}{-4} = -\frac{1}{2} \). Now, we can use the point-slope form of the equation \( y - y_1 = m(x - x_1) \). Plugging in \( m = -\frac{1}{2} \) and the point \( (4, 1) \): \[ y - 1 = -\frac{1}{2}(x - 4) \] Thus, the equation of the line is \( y - 1 = -\frac{1}{2}(x - 4) \), corresponding to option **B**. As for some interesting insights, the relationship between lines and slopes is a fundamental aspect in algebra and geometry. The negative slope we found indicates that as the x-values increase, the y-values decrease, giving you that delightful downhill trend! On applying this information, consider how the slope-intercept form \( y = mx + b \) benefits you in everyday scenarios, such as budgeting or determining speed in travel. The principles of linear equations can guide decision-making and problem-solving in various fields, including economics and engineering!