Which of the following pairs of equations are perpendicular to each other?
3x + y = 9 and y = 4x – 4
x + y = 2 and y = x + 6
5x + 8y = 1 and 2x + 6y = 2
3x – y = 7 and y = 3x + 9
The pair of equations that are perpendicular to each other are:
3x + y = 9 and y = 4x – 4
To determine if two lines are perpendicular to each other, we need to check the slopes of their equations. If the slopes are negative reciprocals of each other, then the lines are perpendicular.
Let's write the equations in slope-intercept form (y = mx + b) to easily determine the slopes.
1) 3x + y = 9 can be rewritten as y = -3x + 9
y = 4x - 4 remains the same
Comparing the slopes:
The slope of the first equation is -3, and the slope of the second equation is 4.
Since -3 and 4 are not negative reciprocals, these lines are not perpendicular.
2) x + y = 2 can be rewritten as y = -x + 2
y = x + 6 remains the same
Comparing the slopes:
The slope of the first equation is -1, and the slope of the second equation is 1.
Since -1 and 1 are negative reciprocals, these lines are perpendicular.
3) 5x + 8y = 1 can be rewritten as y = (-5/8)x + 1/8
2x + 6y = 2 can be rewritten as y = (-1/3)x + 1/3
Comparing the slopes:
The slope of the first equation is -5/8, and the slope of the second equation is -1/3.
Since (-5/8) and (-1/3) are not negative reciprocals, these lines are not perpendicular.
4) 3x - y = 7 can be rewritten as y = 3x - 7
y = 3x + 9 remains the same
Comparing the slopes:
The slope of the first equation is 3, and the slope of the second equation is also 3.
Since both slopes are the same, but not negative reciprocals, these lines are not perpendicular.
Therefore, the only pair of equations that are perpendicular to each other is:
x + y = 2 and y = x + 6.