Are the pairs of lines parallel, perpendicular, or intersecting?
y=−23x+7
−2x−3y=12
parallel, perpendicular, or intersecting?
To determine the relationship between the two lines, we must first write both equations in slope-intercept form (y = mx + b), where m represents the slope of the line.
The first equation is already in slope-intercept form:
y = -23x + 7
The slope of this line is -23.
For the second equation, we need to solve for y to put it into slope-intercept form:
−2x − 3y = 12
Rearrange terms to solve for y:
−3y = 2x + 12
Now divide both sides by -3 to solve for y:
y = (2x + 12) / -3
y = (-2/3)x - 4
The slope of this line is -2/3.
Now that we have the slopes of both lines, we can compare them. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, then the lines are perpendicular. In all other cases, the lines are intersecting.
The slopes of the two lines are -23 and -2/3, respectively. Since these slopes are not equal, the lines are not parallel. They are also not negative reciprocals of each other because the negative reciprocal of -23 would be 1/23, not -2/3. Therefore, the lines are intersecting.
y−6=14(x+2)
y=−4x−10
To determine whether the lines are parallel, perpendicular, or intersecting, we need to express both equations in the slope-intercept form (y = mx + b), where m is the slope.
The first equation is in point-slope form:
y − 6 = 14(x + 2)
We can convert it to slope-intercept form by expanding and simplifying:
y − 6 = 14x + 28
y = 14x + 28 + 6
y = 14x + 34
The slope (m) of this line is 14.
The second equation is already in slope-intercept form:
y = −4x − 10
The slope (m) of this line is -4.
Let's compare the slopes:
The first line has a slope of 14.
The second line has a slope of -4.
Since the slopes are neither equal nor negative reciprocals of each other, the lines are neither parallel nor perpendicular. Therefore, these two lines are intersecting.
To determine whether the pairs of lines are parallel, perpendicular, or intersecting, we can compare their slopes.
First, let's write the equations of the lines in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
For the first line, y = -23x + 7, the slope is -23.
For the second line, -2x - 3y = 12, let's rearrange it into slope-intercept form:
-2x - 3y = 12
-3y = 2x + 12
y = (-2/3)x - 4
From the second equation, we can see that the slope is -2/3.
Now, we compare the slopes:
If the slopes are equal, then the lines are parallel.
If the slopes are negative reciprocals of each other (i.e., the product of their slopes is -1), then the lines are perpendicular.
If the slopes are neither equal nor negative reciprocals, then the lines intersect.
Let's calculate the slopes and determine the relationship between the lines:
Slope of the first line: -23
Slope of the second line: -2/3
Since -23 is not equal to -2/3 and their product is not -1, the lines are neither parallel nor perpendicular. Therefore, the lines intersect.
In summary, the given pair of lines intersect.