Two airplanes mapped their courses on a coordinate grid. The first plane followed a path of y = 3x - 8. The second airplane flew its daily flight on the path -6x + 2y = -16. Describe the similarities or differences in the planes' paths.
To describe the similarities or differences in the planes' paths, we need to analyze their equations:
First plane: y = 3x - 8
Second plane: -6x + 2y = -16
The first plane's equation is in slope-intercept form (y = mx + b), where the coefficient of x represents the slope of the line (3 in this case) and the constant term (b) represents the y-intercept (-8 in this case).
The second plane's equation is in standard form, which can be converted to slope-intercept form by isolating y:
-6x + 2y = -16
2y = 6x - 16
y = 3x - 8/2
Comparing the equations of both planes, we can see that they have the same slope (3), indicating that both planes fly at the same rate vertically relative to their horizontal movement. This means that for every increase of one unit in x, both planes increase their altitude by three units.
Regarding the y-intercepts, the first plane has a y-intercept of -8, while the second plane has a y-intercept of -8/2 = -4. This means that the first plane starts its journey eight units below the y-axis, while the second plane starts four units below the y-axis.
In summary, the similarities in the planes' paths are that they have the same slope (3) and both start below the y-axis. The difference lies in their y-intercepts; the first plane starts deeper below the y-axis compared to the second plane.