Answer:
1. C: -3/4x + 3
2. E: 3/4x - 1
3. I: -2/5x + 3
4. A: 2/5x - 1
Explanation:
(1. )
Importance of the slope (m) in parallel lines
- Before we can find the equation of the line that is parallel to the graphed line, we need to know the slope (m) of the line.
- We need to know the slope since the slopes of parallel lines are the same.
Finding the slope (m):
We can find the slope (m) of the line using the slope formula, which is given by:
m = (y2 - y1) / (x2 - x1), where
- m is the slope,
- (x1, y1) is one point on the line,
- and (x2, y2) is another point.
Since (-4, 4) and (0, 1) lie on the line, we can find the slope (m) by substituting (-4, 4) for (x1, y1) and (0, 1) for (x2, y2) in the slope formula:
m = (1 - 4) / (0 - (-4))
m = (-3) / (0 + 4)
m = -3/4
Thus, the slope is -3/4.
Determining the equation of the line parallel to the graphed line:
Since the equation for answer choice C. is y = -3/4x + 3 and -3/4 is the slope, this is the equation of a line parallel to the line in 1.
----------------------------------------------------------------------------------------------------------
(2.)
Importance of the slope in perpendicular liens
- We'll also need to know the slope (m) of 4x + 3y = 12 before we can find the equation of the line perpendicular to it.
- We need to first find the slope (m) since the slopes of perpendicular lines are negative reciprocals of each other.
This is shown by the formula:
m2 = -1 / m1, where
- m2 is the slope of the other line,
- and m1 is the slope of the line we're given (or we can easily find it, which is the case with 4x + 3y = 12).
Converting from standard form to slope-intercept form and identifying the slope (m):
4x + 3y = 12 is in the standard form of a line, whose general equation is given by:
Ax + By = 12
We can easily find the slope (m) by converting from standard form to slope-intercept form, whose general equation is given by:
y = mx + b, where
- m is the slope,
- and b is the y-intercept.
Thus, we convert from standard to slope-intercept form by isolating y on the left-hand side:
(4x + 3y = 12) - 4x
(3y = -4x + 12) / 3
y = -4/3x + 4
Thus, the slope (m) of 4x + 3y = 12 is -4/3.
Determining the equation of the line perpendicular to 4x + 3y = 12:
Now we can plug in -4/3 for m1 in the perpendicular slope formula above to find m2, the slope of the other line (knowing m2 will also tells us which equation is perpendicular to 4x + 3y = 12):
m2 = -1 / (-4/3)
m2 = -1 * -3/4
m2 = 3/4
Thus, the slope of the other line perpendicular to 4x + 3y = 12 is 3/4.
Since the equation for answer choice E. is y = 3/4x - 1 and 3/4 is its slope, this is the equation of the line perpendicular to 4x + 3y = 12.
----------------------------------------------------------------------------------------------------------
(3.)
Importance of the slope in parallel lines:
- Before we find the equation of the line parallel to 2x + 5y = 10, we'll need to know its slope (m).
- We already know that the slopes (m) of parallel lines are the same.
Converting form standard form to slope-intercept form and identifying the slope:
- We know that 2x + 5y = 10 is in the standard form of a line.
We can convert from standard form (i.e., Ax + By = C) to slope-intercept form (i.e., y = mx + b) by isolating y on the left-hand side:
(2x + 5y = 10) - 2x
(5y = -2x + 10) / 5
y = -2/5x + 2
Thus, the slope (m) of 2x + 5y = 10 is -2/5.
Determining the equation of the line parallel to 2x + 5y = 10:
Since the equation for answer choice I. is y = -2/5x + 3 and -2/5 is its slope, this is the equation of the line parallel to 2x + 5y = 10.
----------------------------------------------------------------------------------------------------------
(4.)
Importance of the slope (m) in perpendicular lines:
- Before we can find the equation of the line perpendicular to the graphed line, we'll need to know its slope.
Finding the slope (m) of the graphed line:
Like we did for 1., we can find the slope (m) of the line using the slope formula (i.e., m = (y2 - y1) / (x2 - x1))
Since (0, 5) and (2, 0) are points on the line, we can find the slope (m) by substituting (0, 5) for (x1, y1) and (2, 0) for (x2, y2) in the slope formula:
m = (0 - 5) / (2 - 0)
m = -5/2
Thus, the slope of the graphed line is -5/2.
Determining the equation of the line perpendicular to the graphed line:
- We can again use the perpendicular slope formula (m2 = -1 / m1) to determine the equation of the line perpendicular to the graphed line using its slope.
To find the slope of the other line, we substitute -5/2 for m1 in the perpendicular slope formula:
m2 = -1 / (-5/2)
m2 = -1 * -2/5
m2 = 2/5
Thus, the slope of the other line is 2/5.
Since the equation for answer choice A. is y = 2/5x - 1 where 2/5 is the slope, this is the equation of the line perpendicular to the graphed line: