Find the equation (in terms of \( x \) ) of the line through the points \( (-1,-4) \) and \( (4,4) \) \( y= \)

To find the equation of the line through the points \((-1,-4)\) and \((4,4)\), we first need to calculate the slope \(m\) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, \(y_2 = 4\), \(y_1 = -4\), \(x_2 = 4\), and \(x_1 = -1\). Plugging in these values, we get: \[ m = \frac{4 - (-4)}{4 - (-1)} = \frac{4 + 4}{4 + 1} = \frac{8}{5}. \] Now we use the point-slope form of the line’s equation, which is \(y - y_1 = m(x - x_1)\). Using the point \((-1,-4)\): \[ y - (-4) = \frac{8}{5}(x - (-1)), \] \[ y + 4 = \frac{8}{5}(x + 1). \] Distributing the slope: \[ y + 4 = \frac{8}{5}x + \frac{8}{5}. \] Now, isolating \(y\): \[ y = \frac{8}{5}x + \frac{8}{5} - 4, \] \[ y = \frac{8}{5}x + \frac{8}{5} - \frac{20}{5}, \] \[ y = \frac{8}{5}x - \frac{12}{5}. \] Therefore, the equation of the line in terms of \(x\) is: \[ y = \frac{8}{5}x - \frac{12}{5}. \] This gives you the relationship between \(x\) and \(y\) along that straight path connecting those two points! Now, let’s dive into some history and practical applications of lines! Did you know that the concept of a straight line dates back to ancient civilizations? The earliest contributions can be attributed to the Greeks, particularly Euclid, who introduced the idea of geometric properties and constructed various theories surrounding lines and planes. This foundation set the stage for modern mathematics! In the real world, linear equations are everywhere! For instance, in economics, the demand and supply curves can be modeled using linear equations to predict how price changes affect the quantity of goods sold. Understanding the slope and y-intercept helps businesses make decisions based on these predictions.