What is an equation of the line that passes through the points \( (4,-2) \) and \( (6,1) \) ?

To find the equation of the line passing through the points \( (4,-2) \) and \( (6,1) \), we first need to calculate the slope (m) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-2)}{6 - 4} = \frac{3}{2} \] With the slope \( m = \frac{3}{2} \) and the point \( (4, -2) \), we can use the point-slope form of the line's equation: \[ y - y_1 = m(x - x_1) \] Substituting the values, we get: \[ y - (-2) = \frac{3}{2}(x - 4) \] This simplifies to: \[ y + 2 = \frac{3}{2}x - 6 \] Thus, the equation becomes: \[ y = \frac{3}{2}x - 8 \] Now you have the equation of the line! Did you know that in the history of mathematics, the concept of a line and its equation has been around for thousands of years? Ancient Egyptians and Greeks were already discussing geometric properties, laying the groundwork for our current understanding. The straight-line equation as we know it today began taking shape during the Renaissance, thanks to mathematicians like Descartes and Fermat, who melded algebra with geometry. If you're keen on diving deeper into linear equations, exploring the wondrous world of graphing is essential! Check out interactive graphing tools or apps like Desmos. They make visualizing these equations a breeze and provide an excellent way to experiment with slopes and intercepts!