Final answer:
To find the equation of the line through (-2, 8) and (1, -2), calculate the slope which is -10/3. Then, use the point-slope form to get y - 8 = (-10/3)(x + 2), and simplify to the slope-intercept form, resulting in y = (-10/3)x + 4/3, representing the line equation.
Step-by-step explanation:
Finding the Equation of a Line
To find the equation of a line that passes through two given points, (-2, 8) and (1, -2), we first need to calculate the slope (m) of the line. The slope is the change in y divided by the change in x. Using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are points on the line, we can compute the slope:
m = (-2 - 8) / (1 - (-2)) = -10 / 3 = -10/3
With the slope and one of the points, say (-2, 8), we can use the point-slope form of the equation y - y1 = m(x - x1) to find the equation of the line:
y - 8 = (-10/3)(x - (-2))
y - 8 = (-10/3)(x + 2)
Distributing the slope on the right side:
y - 8 = (-10/3)x - (20/3)
To convert to slope-intercept form, we add 8 to both sides:
y = (-10/3)x - (20/3) + 24/3
y = (-10/3)x + 4/3
This is the equation of the line in slope-intercept form, where -10/3 is the slope and 4/3 is the y-intercept.
Understanding linear equations is crucial because they can be applied in various contexts, including statistics, economics, and engineering. For example, a regression line is a type of linear equation used in statistics to predict values. The line of best fit, for example, uses the least-squares method to minimize the distances of all points from the line, resulting in a predictive model which can forecast new data points given past trends.