Compare the following two equations with additive relationships: x=3y−12 and y=−3x+4 . Determine whether the lines of these equations are parallel or perpendicular, if they go through the origin, and if they share a y -intercept. Show your work. in 7th grader words
To compare the equations x=3y−12 and y=−3x+4, let's first rewrite them in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
For the equation x=3y−12, we need to isolate y:
x + 12 = 3y (add 12 to both sides)
(x + 12)/3 = y (divide both sides by 3)
y = (1/3)x + 4 (slope-intercept form equation)
For the equation y=−3x+4, the equation is already in slope-intercept form.
Now, let's compare the slopes (m) of the two equations. The slope of the first equation is 1/3, while the slope of the second equation is -3. Since the slopes are not the same, the lines are not parallel.
To determine if the lines are perpendicular, we can check if the product of their slopes is -1. In this case, (1/3) * (-3) = -1, which means the lines are perpendicular.
To check if the lines go through the origin, we can substitute (0,0) into both equations and see if the equations hold true. For the first equation, when x=0, y should also equal 0:
0 = (1/3) * 0 + 4
0 = 0 + 4
0 = 4 (This is not true)
For the second equation, when y=0, x should also equal 0:
0 = -3 * 0 + 4
0 = 0 + 4
0 = 4 (This is not true)
Therefore, neither equation goes through the origin.
To check if they share a y-intercept, we can compare their y-intercept values. The y-intercept of the first equation is 4, and the y-intercept of the second equation is also 4. Since they have the same y-intercept value, the lines share a y-intercept.
Overall, the lines of the two equations are neither parallel nor perpendicular, they do not go through the origin, but they do share a y-intercept.