ompare 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.(4 points)
To compare the two equations for additive relationships, we can rewrite them in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
1. Equation 1: x = 3y - 12
Rearranging the equation to slope-intercept form, we get:
3y = x + 12
y = (1/3)x + 4
So the slope of this line is 1/3 and the y-intercept is 4.
2. Equation 2: y = -3x + 4
This equation is already in slope-intercept form, where the slope is -3 and the y-intercept is 4.
Now, let's analyze the relationship between the lines based on their slopes:
- If the slopes are equal, the lines are parallel.
- If the slopes are negative reciprocals, the lines are perpendicular.
Comparing the slopes:
- The slope of Equation 1 is 1/3.
- The slope of Equation 2 is -3.
The slopes are negative reciprocals of each other (-3 * 1/3 = -1), so the lines are perpendicular.
Next, let's check if they go through the origin:
- To test if a line goes through the origin, we substitute x = 0 and y = 0 into the equation.
- Equation 1: x = 3y - 12
If we plug in x = 0, we get: 0 = 3y - 12
Solving for y, we get: y = 4
So the line represented by Equation 1 does not go through the origin.
- Equation 2: y = -3x + 4
If we plug in x = 0, we get: y = -3(0) + 4
Solving for y, we get: y = 4
So the line represented by Equation 2 does go through the origin.
Finally, let's check if they share a y-intercept:
- The y-intercept of Equation 1 is 4.
- The y-intercept of Equation 2 is also 4.
The two equations share the same y-intercept.
To summarize:
- The lines represented by the equations are perpendicular.
- Equation 1 does not go through the origin, while Equation 2 does.
- The two equations share the same y-intercept.