Write an equation of the line that is:

a) Parallel to the line \(y = 2x - 4\), and has a y-intercept of 7.

b) Parallel to the line \(y - 3x = 6\), and has a y-intercept of -2.

c) Parallel to the line \(2x + 3y = 12\), and passes through the origin.

d) Perpendicular to the line \(y = 3x + 2\), and has a y-intercept of 2.

e) Perpendicular to the line \(3y + 4x = 18\), and passes through the origin.

Please answer all these questions.

Lines that are parallel will have the same slope. Lines that are perpendicular will have slopes that are the negative reciprocal of each other.
*(a) * parallel to the line y=2x-4, and has a y-intercept of 7 slope of first line: 2 slope of second line: 2 y=mx + b y=2x+7 (b) parallel to the line y-3x=6, and has a y-intercept of -2 y=3x+6 slope of first line: 3 slope of second line: 3 y=mx + b y=3x-2
*(c) * parallel to the line 2x+3y=12, and that passes through the origin 3y=-2x+12 y=-2/3x+4 slope of first line: -2/3 slope of second line: -2/3 origin: (0,0) y=mx + b y=-2/3x + 0 y=-2/3x
*(d) * perpendicular to the line y=3x+2, and has a y-intercept of 2 slope of first line: 3 slope of second line: -1/3 y=mx + b y=-1/3x + 2
*(e) * perpendicular to the line 3y+4x=18, and that passes through the origin 3y=4x+18 y=4/3x+6 slope of first line: 4/3 slope of second line: -3/4 origin: (0,0) y=mx + b y=-3/4x + 0 y=-3/4x

Answered by iloveonedirection | 2024-06-10

(a)Equation of line is: y = 2x + 7. (b)Equation: y = 3x - 2 (c)Equation: y = -2/3x.(d)Equation: y = -1/3x + 2.(e)Equation: y = 3/4x.
We'll solve each part step by step.

Part (a)

To find a line parallel to y = 2x - 4 with a y-intercept of 7, we keep the same slope (2) since parallel lines have identical slopes.
Equation: y = 2x + 7

Part (b)

The given line is y - 3x = 6. Rewrite it in slope-intercept form (y = mx + b) to find the slope:
y = 3x + 6. The slope (m) is 3. A parallel line with a y-intercept of -2 will be:
Equation: y = 3x - 2

Part (c)

First, convert 2x + 3y = 12 to slope-intercept form:
3y = -2x + 12,
y = -2/3x + 4. The slope is -2/3. A line parallel and passing through the origin will be:
Equation: y = -2/3x

Part (d)

The given line is y = 3x + 2. Perpendicular lines have slopes that are negative reciprocals. Thus, the slope will be -1/3:
Equation: y = -1/3x + 2

Part (e)

Convert 3y + 4x = 18 to slope-intercept form:
3y = -4x + 18,
y = -4/3x + 6. The slope is -4/3. A perpendicular line's slope will be the negative reciprocal, 3/4:
Equation: y = 3/4x

Answered by Qwballon | 2024-06-26

The equations of the specified lines are: 1) y = 2 x + 7 , 2) y = 3 x − 2 , 3) y = − 3 2 ​ x , 4) y = − 3 1 ​ x + 2 , and 5) y = 4 3 ​ x . Each equation is derived based on the slope of the reference line and the specified y-intercepts or conditions. Understanding the concepts of slope and y-intercept is key to solving such problems.
;

Answered by iloveonedirection | 2024-12-24