Solutions to Line and Slope Problems

Solutions to the line problems are presented along with detailed explanations .

Solution to Problem 1:
The slope of the line is given by
m = (y2 - y1) / (x2 - x1) = (12 - 0) / (-4 - (-1)) = - 12 / 3 = - 4
We now write the equation of the line in point slope form: y - y1 = m (x - x1)
y - 0 = - 4(x - (-1))
Simplify and write the equation in general form
y + 4x = - 4

Solution to Problem 2:
The two points have the same x coordinate and are on the same vertical line whose equation is
x = - 2

Solution to Problem 3:
The two points have the same y coordinate and are on the same horizontal line whose equation is
y = 5

Solution to Problem 4:
A line parallel to the axis has equation of the form y = constant. Since the line we are trying to find passes through (3 , 4), then the equation of the line is given by:
y = 4

Solution to Problem 5:
The x intercept is the point (-1 , 0). The slope of the line is given by:
m = (2 - 0) / (-3 - (-1)) = 2 / - 2 = -1
The point slope form of the line is
y - 0 = -1(x - (-1))
The equation can be written as
y = - x - 1

Solution to Problem 6:
The x and y intercepts are the points (-4 , 0) and (0 , 5). The slope of the line is given by:
m = (5 - 0) / (0 - (-4)) = 5 / 4
The point slope form of the line is
y - 5 = (5 / 4)(x - 0)
Multiply all terms by 4 and simplify
4 y - 20 = 5 x

Solution to Problem 7:
The line y = 9 is a horizontal line (parallel to the x axis). The line that is perpendicular to the line y = 9 have the form x = constant. Since the (-1 , 0) is a point on this line, the equation is given by
x = -1

Solution to Problem 8:
To find the slope of the given, we first write in slope intercept form
5y = 3x + 8
y = (3/5)x + 8/5
The slope is equal to 3/5. The y intercept is found by setting x = 0 in the equation and solve for y. Hence the y intercept is at y = 8/5. The x intercept is found by setting y = 0 and solve for x. Hence the x intercept is at x = -8/3

Solution to Problem 9:
Given the equation
x / 4 - y / 5 = 3
Keep only the term in y on the left side of the equation
- y / 5 = 3 - x / 4
Multiply all terms by -5
y = (5/4)x - 15

Solution to Problem 10:
The line x = - 3 is parallel to the y axis and the line x = 0 is the y axis. The two lines are parallel.

Solution to Problem 11:
For a point to be on a line, its coordinates must satisfy the equation of the line.
2 - 4(2b) = 6
Solve for b
b = -1/2

More math problems with detailed solutions in this site.

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