Find an equation of a line passing through the points \( (4,10) \) and \( (12,1) \cdot y=\square \)

To find the equation of the line passing through the points \( (4,10) \) and \( (12,1) \), we first calculate the slope (m) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1) = (4, 10) \) and \( (x_2, y_2) = (12, 1) \). Calculating the slope: \[ m = \frac{1 - 10}{12 - 4} = \frac{-9}{8} \] Now, we can use the point-slope form of the equation of a line, which is \( y - y_1 = m(x - x_1) \). Using the point \( (4, 10) \): \[ y - 10 = \frac{-9}{8}(x - 4) \] Now, rearranging into slope-intercept form (y = mx + b): \[ y - 10 = \frac{-9}{8}x + \frac{36}{8} \] \[ y = \frac{-9}{8}x + \frac{36}{8} + 10 \] \[ y = \frac{-9}{8}x + \frac{36 + 80}{8} \] \[ y = \frac{-9}{8}x + \frac{116}{8} \] Thus, the equation of the line is: \[ y = \frac{-9}{8}x + \frac{29}{2} \] --- Isn't it fascinating that the concept of slopes can be traced back to ancient civilizations? Early mathematicians like the Greeks and Indians explored ratios and proportions, which laid the groundwork for understanding linear relationships! Real-world applications of linear equations are everywhere! For instance, in economics, the relationship between supply and demand can often be modeled with a linear equation. Companies use these equations to predict how changes in price might affect sales, making it a crucial tool in planning and strategy!