1. Tell whether the lines for the pair of equations are parallel, perpendicular, or neither. Y=-3/4x-2
A. Parallel
B. Perpendicular*
C. Neither
2. Write the equation of a line that is perpendicular to the given line and that passes through the given point. Y-3=-1/5 (x+2); (-2, 7)
A. y=5x+7
B. y=5+17
C. y=(1/5)x-2
D. y=-2x+3
1. To determine whether two lines are parallel, perpendicular, or neither, we need to compare their slopes. The given equation is written in slope-intercept form, which is y = mx + b, where m represents the slope of the line.
In the given equation y = -3/4x - 2, the slope is -3/4.
If two lines are perpendicular, their slopes are negative reciprocals of each other. The negative reciprocal of -3/4 is 4/3.
Therefore, since the slope of the given equation is -3/4 and the negative reciprocal of that is 4/3, the lines are perpendicular.
The correct answer is B. Perpendicular.
2. To find the equation of a line that is perpendicular to the given line and passes through the given point, we need to determine the slope of the given line and then find its negative reciprocal.
The given equation is y - 3 = -1/5 (x + 2). To find the slope of this line, we can rewrite the equation in slope-intercept form.
y - 3 = -1/5 (x + 2)
y - 3 = -1/5x - 2/5
y = -1/5x - 2/5 + 3
y = -1/5x - 2/5 + 15/5
y = -1/5x + 13/5
The slope of the given line is -1/5.
To find the slope of the line perpendicular to this, we take the negative reciprocal of -1/5. The negative reciprocal of -1/5 is 5/1 or simply 5.
Now we have the slope of the perpendicular line, which is 5, and the point through which it passes, which is (-2, 7). We can use the point-slope form of the equation, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Plugging in the values, we get:
y - 7 = 5(x + 2)
Simplifying this, we have:
y - 7 = 5x + 10
y = 5x + 17
Therefore, the equation of the line that is perpendicular to the given line and passes through the given point (-2, 7) is y = 5x + 17.
The correct answer is B. y = 5x + 17.
To determine the relationship between two lines, we need to analyze their slopes. The slope-intercept form of a line equation is y = mx + b, where m represents the slope.
1. For the given equation y = -3/4x - 2, we can identify that the slope is -3/4.
To determine the relationship between lines, we compare their slopes.
- If two lines have the same slope, they are parallel.
- If two lines have slopes that are negative reciprocals of each other (i.e., their slopes multiply to -1), they are perpendicular.
- If the slopes are neither equal nor negative reciprocals, then the lines are neither parallel nor perpendicular.
Comparing the slope of y = -3/4x - 2 to the slope of a line perpendicular to it, we need to find the negative reciprocal of -3/4. The negative reciprocal of -3/4 is 4/3.
Therefore, the lines are perpendicular. The answer is B. Perpendicular.
2. Given the line equation y - 3 = -1/5 (x + 2) and a point (-2, 7), we need to find the equation of a line that is perpendicular to the given line and passes through the given point.
We can start by rearranging the given equation into slope-intercept form:
y - 3 = -1/5 (x + 2)
y - 3 = -1/5x - 2/5
y = -1/5x + (3 - 2/5)
y = -1/5x + 13/5
Now comparing this equation with the standard form y = mx + b, we can see that the slope of the given line is -1/5.
To find the slope of the line perpendicular to it, we need to calculate the negative reciprocal of -1/5, which is 5/1 or simply 5.
Using the point-slope form of a line equation (y - y1 = m(x - x1)) and plugging in the point (-2, 7), we can find the equation of the perpendicular line:
y - 7 = 5(x - (-2))
y - 7 = 5(x + 2)
y - 7 = 5x + 10
y = 5x + 17
Therefore, the equation of the line perpendicular to the given line and passing through the point (-2, 7) is y = 5x + 17. The answer is B. y = 5x + 17.
Do you know at least to write a question?
1.
pair of equations
y = - 3 / 4 x - 2
What is the other equation?
2.
y = 5 + 17 ???
2.
y - 3 = - 1 / 5 ( x + 2 )
y - 3 = - 1 / 5 x + 2 / 5
Add 3 to both sides
y - 3 + 3 = - 1 / 5 x + 2 / 5 + 3
y = - 1 / 5 x + 2 / 5 + 3
y = - 1 / 5 x + 2 / 5 + 15 / 5
y = - 1 / 5 x + 17 / 5
y = m x + b
y = - 1 / 5 x + 17 / 5
m = - 1 / 5
A perpendicular line has a slope that is the negative reciprocal of this.
mp = - 1 / m = - 1 / ( - 1 / 5 ) = 5
You need the equation of the line through
( - 2 , 7 ) which means x = - 2 , y = 7
with slope 5
y = mp x + b
7 = 5 ∙ ( - 2 ) + b
7 = - 10 + b
Add 10 to both sides
7 + 10 = - 10 + b + 10
17 = b
b = 17
y = m x + b
y = 5 x + 17