Find an equation of the line having the given slope and containing the given point. \( \mathrm{m}=\frac{6}{7},(4,-1) \)

To find the equation of the line in slope-intercept form (y = mx + b), we can start by using the given slope \( m = \frac{6}{7} \) and the point \( (4, -1) \). First, we plug in the point into the point-slope formula: \[ y - y_1 = m(x - x_1) \] Substituting \( m = \frac{6}{7} \), \( x_1 = 4 \), and \( y_1 = -1 \): \[ y - (-1) = \frac{6}{7}(x - 4) \] Solving this gives: \[ y + 1 = \frac{6}{7}x - \frac{6}{7} \cdot 4 \] \[ y + 1 = \frac{6}{7}x - \frac{24}{7} \] \[ y = \frac{6}{7}x - \frac{24}{7} - 1 \] \[ y = \frac{6}{7}x - \frac{24}{7} - \frac{7}{7} \] \[ y = \frac{6}{7}x - \frac{31}{7} \] So the equation of the line is: \[ y = \frac{6}{7}x - \frac{31}{7} \] --- Let’s take a brief look at how slopes work in the world around us! Slopes are everywhere—from the steep hills we climb to the gentle ramps for wheelchairs. Engineers use slope measurements to design everything from roads to roofs, ensuring safety and functionality. The concept of slope gives us insight into how quickly one variable changes in relation to another. Also, if you’re into exploring the world of lines, there are loads of resources to dive deeper! You might check out books on algebra where they explore linear equations or even websites with interactive graphing tools. Whether you prefer learning through videos or hands-on activities, there’s a whole universe waiting for you to discover and play with slopes, intercepts, and lines!