 # Equations of Lines: Problems with Solutions

Find equations and slope of lines; questions and problems with solutions are presented.

## Problems

Problem 1: Find the equation of the line that passes through the points (-1 , 0) and (-4 , 12).

Problem 2: What is the equation of the line through the points (-2 , 0) and (-2 , 4).

Problem 3: Find the equation of the line that passes through the points (7 , 5) and (-9 , 5).

Problem 4: Find the equation of the line through the point (3 , 4) and parallel to the x axis.

Problem 5: What is the equation of the line through the point (-3 , 2) and has x intercept at x = -1.

Problem 6: Find the equation of the line that has an x intercept at x = - 4 and y intercept at y = 5.

Problem 7: What is the equation of the line through the point (-1 , 0) and perpendicular to the line y = 9.

Problem 8: Find the slope, the x and y intercepts of the line given by the equation: -3 x + 5 y = 8.

Problem 9: Find the slope intercept form for the line given by its equation: x / 4 - y / 5 = 3.

Problem 10: Are the lines x = -3 and x = 0 parallel or perpendicular?

Problem 11: For what values of b the point (2 , 2 b) is on the line with equation x - 4 y = 6

## Solutions to the Above Problems

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 + 4 x = - 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(2 b) = 6
Solve for b
b = -1 / 2