Equation of Line Questions with Solutions

Question on how to find the slopes and equations of lines. A review of the concepts of slope and equations of lines are presented followed by questions with detailed solutions.

Slopes and Equations of Lines

Slope of a Line:

If a line passes through two distinct points P₁(x₁ , y₁) and P₂(x₂, y₂), its slope is given by:

m = (y₂ - y₁) / (x₂ - x₁)

with x₂ not equal to x₁.

General Equation of a Straight line:

The general equation of straight line is given by:

A x + B y = C

where A, B and C are constants and A and B cannot be both zero. An interactive exploration of the equations of lines of the form A x + B y = c is included.
Any straight line in a rectangular system has an equation of the form given above.

Slope intercept form of a Line:

The equation of a line with a defined slope m can also be written as follows:

y = m x + b

where m is the slope of the line and b is the y intercept of the graph of the line.
The above form is called the slope intercept form of a line. Further interactive tutorials on this form of lines are included.

Point-Slope form of a line:

An equation of a line through a point P(x₁ , y₁) with slope m is given by

y - y₁ = m(x - x₁)

Vertical and Horizontal lines:

a - If we set A to zero in the general equation, we obtain an equation in y only of the form

B y = C

which gives y = C / B = k; k is a constant. This is a horizontal line with slope 0 and passes through all points with y coordinate equal to k = C / B.

b - If we set B to zero in the general equation, we obtain

A x = C

which gives x = C / A = h; h is constant. This is a vertical line with undefined slope and passes through all points with x coordinate equal to h.

Parallel Lines:

Two non vertical lines are parallel if and only if their slopes are equal.

Perpendicular Lines:

Two non vertical lines are perpendicular if and only if their slopes m₁ and m₂ are such that

m₁ × m₂ = - 1

Questions with Solutions

Question 1
Find the slope of a line passing through the points

(2 , 3) and (0 , - 1)
(- 2 , 4) and (- 2 , 6)
(5 , 2) and (- 7 , 2)

Solution to Question 1

m = (y₂ - y₁) / (x₂ - x₁) = (-1 - 3) / (0 - 2) = 2
m = (6 - 4) / (-2 + 2)
The division by -2 + 2 = 0 is undefined and the slope in this case is undefined. The line passing through the given points is a vertical line.
m = (2 - 2) / (-7 - 5) = 0
The slope is equal to 0 and the line through the given points is a horizontal line.

Question 2
Find the equation of the line that passes through the point (-2 , 5) and has a slope of -4.
Solution to Question 2

Substitute y₁ , x₁ and m in the point slope form of a line
y - y₁ = m(x - x₁)
y - 5 = - 4(x - (-2))
y = - 4 x - 3

Question 3
Find the equation of the line that passes through the points (0 , -1) and (3 , 5).
Solution to Question 3

We first calculate the slope of the line
m = (5 - (-1)) / (3 - 0) = 6 / 3 = 2
Use the slope and any of the two points to write the equation of the line using the point slope form.
y - y₁ = m(x - x₁)
using the first point
y - (-1) = 2(x - 0)
y = 2 x - 1

Question 4
Find the slope of the line given by the equation

- 2 x + 4 y = 6

Solution to Question 4

Given the equation
- 2 x + 4 y = 6
Write the equation in slope intercept form
4 y = 2 x + 6
y = (1 / 2) x + 3 / 2
The slope of the line is given by the coefficient of x and is equal to 1 / 2.

Question 5
Find an equation of the line that passes through the point (-2 , 3) and is parallel to the line 4 x + 4 y = 8
Solution to Question 5

Let m₁ be the slope of the line whose equation is to be found and m₂ the slope of the given line. Rewrite the given equation in slope intercept form and find its slope.
4 y = - 4 x + 8
Divide both sides by 4
y = - x + 2
slope m₂ = - 1.
Two lines are parallel if and only if they have equal slopes
m₁ = m₂ = - 1
We now use the point slope form to find the equation of the line with slope m₁.
y - 3 = - 1(x - (-2))
which may be written as
y = - x + 1

Question 6
Find an equation of the line that passes through the point (0 , - 3) and is perpendicular to the line - x + y = 2.
Solution to Question 6

Let m₁ be the slope of the line whose equation is to be found and m₂ the slope of the given line. Rewrite the given equation in slope intercept form and find its slope.
y = x + 2
slope m₂ = 1
Two lines are perpendicular if and only their slopes are such that
m₁ × m₂ = - 1
This gives m₁ = -1
We now use the point slope form to find the equation of the line with slope m₁.
y - (-3) = -1(x - 0)
which may be written
y = - x - 3

Matched Questions

The following are questions matched to the questions presented above. Their answers are also included.

Matched Question 1
Find the slope of a line passing through the points

(- 2 , 7) and (- 2 , - 1)
(2 , 4) and (- 2 , 6)
(- 1 , - 2) and (4 , - 2)

Matched Question 2
line that passes through the point (3 , 0) and has a slope of - 1.

Matched Question 3
Find the equation of the line that passes through the points (2 , 0) and (3 , 3).

Matched Question 4
Find the slope of the line given by the equation

x - 3 y = - 9

Matched Question 5
Find an equation of the line that passes through the point (-1 , 0) and is parallel to the line - 2 x + 2 y = 8.

Matched Question 6
Find an equation of the line that passes through the point (-2 , 1) and is perpendicular to the line x + 2 y = -2.

Answers to the Matched Questions

Matched Question 1
1) undefined
2) - 1 / 2
3) 0

Matched Question 2
y = - x + 3

Matched Question 3
y = 3 x - 6

Matched Question 4
slope = 1 / 3

Matched Question 5
y = x + 1

Matched Question 6
y = 2 x + 5