G1 Find slope from two points and write a line equation

Ontario Grade 10 Mathematics

Analytic Geometry

Use two points to describe a line’s change and write a rule for it.

A line on a graph shows a steady relationship between two values. For example, a graph of distance against time may be a straight line when an object moves at a steady rate. Slope tells us how much one value changes compared with another. You will use two points to find a line’s slope, then use the slope and a point to write an equation for the line.

What you will learn

Find the slope of a line through two points.
Explain slope as vertical change compared with horizontal change.
Use the slope and a point to write a line equation in the form $y=mx+b$ .
Recognize a vertical line and write its equation.

1. Grade 9 bridge: coordinates and change

A point on a coordinate grid is written as an ordered pair, such as

(2,1)

. The first number is the

x

-coordinate, which tells the horizontal position. The second number is the

y

-coordinate, which tells the vertical position. The

x

-axis runs across the grid, and the

y

-axis runs up and down.

To compare two points, subtract matching coordinates. The change in

x

is found using the two

x

-coordinates. The change in

y

is found using the two

y

-coordinates. Keep the point order consistent. If you subtract the first point’s coordinate from the second point’s coordinate for

x

, do the same for

y

.

Slope describes vertical change compared with horizontal change. It is also called rise over run. Rise means the change in

y

; run means the change in

x

. A line with positive slope rises as you move from left to right. A line with negative slope falls as you move from left to right. A horizontal line has no vertical change, so its slope is zero.

The letter

m

is commonly used for slope. Slope is the change in

y

divided by the change in

x

. This tells us how much the line moves up or down for each unit it moves sideways.

Subtract $x$ -coordinates from each other and $y$ -coordinates from each other.
Use the same point order for both subtractions.
Slope compares vertical change with horizontal change.

2. See rise and run, then use the slope rule

Imagine moving along a line from one point to another. Count the horizontal movement first. This is the run. Then count the vertical movement. This is the rise. For instance, moving from

(1,2)

to

(3,6)

means moving

2

units right and

4

units up. The slope is

4

divided by

2

, which is

2

. The line rises

2

units for each unit it moves right.

The slope formula uses the coordinates of two points. In the formula, the subscripts

1

and

2

identify the first and second points. You may choose either point as the first point. If you reverse the order, both coordinate differences change signs, so their quotient stays the same. Reversing the order in only one subtraction gives an incorrect slope.

A run of zero means the line does not move sideways between the points. Dividing by zero is not possible, so the slope is undefined. This happens for a vertical line. A vertical line has the same

x

-coordinate at every point.

m=\frac{y_2-y_1}{x_2-x_1}

Run is the horizontal change; rise is the vertical change.
Slope is rise divided by run.
A vertical line has zero run and an undefined slope.

3. Use the slope and a point to write an equation

An equation of a line is a rule that is true for every point on the line. A useful form is

y=mx+b

. The letter

m

stands for the slope. The letter

b

is the

y

-intercept, which is the

y

-coordinate where the line crosses the

y

-axis.

Once you know the slope, substitute the coordinates of a point on the line into the equation. The known

x

- and

y

-values and the slope let you find

b

. Then write the equation using the slope and the value of

b

. Either point from the question can be used.

Check your equation by substituting the coordinates of a point into it. If the equation is correct, its left and right sides will be equal. When two points are given, checking both points is a useful way to catch a calculation or substitution error.

y=mx+b

In $y=mx+b$ , $m$ is the slope and $b$ is the $y$ -intercept.
Substitute a known point to find $b$ .
Check the equation by substituting the given points.

4. Special lines and independent practice

A vertical line has the same

x

-coordinate at every point. Its run is zero, so it has an undefined slope. Write its equation by stating the fixed

x

-coordinate. For example, if every point has an

x

-coordinate of

4

, its equation is

x=4

.

A horizontal line has the same

y

-coordinate at every point. Its rise is zero, so its slope is zero. Its equation states the fixed

y

-coordinate. These cases help you decide whether a slope calculation makes sense.

Independent practice: Find the slope through

(1,5)

and

(4,-1)

, then write the equation in the form

y=mx+b

. Keep the subtraction order consistent. Use one point to find

b

, then substitute both points into your equation to check it.

A vertical line has zero run, an undefined slope, and an equation such as $x=4$ .
A horizontal line has zero rise and slope zero.
Substituting the given points checks whether your equation fits the line.

Reading rise and run from two points

Starting point	Ending point	Run: change in $x$	Rise: change in $y$	Slope
$(1,2)$	$(3,6)$	$3-1=2$	$6-2=4$	$4/2=2$
$(2,1)$	$(6,9)$	$6-2=4$	$9-1=8$	$8/4=2$

Worked example

Find the slope and equation

Find the slope of the line through

(2,1)

and

(6,9)

. Then write its equation in the form

y=mx+b

.

Find the coordinate changes
Use $(2,1)$ as the first point and $(6,9)$ as the second point. Subtract the first point’s coordinates from the second point’s matching coordinates.
$\Delta x=6-2=4,\quad \Delta y=9-1=8$
Calculate the slope
Divide the change in $y$ by the change in $x$ . The line rises $8$ units while moving $4$ units to the right.
$m=\frac{8}{4}=2$
Find the intercept
Substitute the slope and the point $(2,1)$ into $y=mx+b$ . Then solve for $b$ .
$1=2(2)+b,\quad b=-3$
Write and check the equation
Place the slope and intercept in the line form. Substitute the second point to check that the equation is true for it.
$y=2x-3,\quad 9=2(6)-3$

Answer: The slope is

2

, and the equation is

y=2x-3

.

Check: For the first point, the equation gives

1=2(2)-3

. For the second point, it gives

9=2(6)-3

. Both points satisfy the equation.

Common mistakes and how to avoid them

Using an

x

-coordinate and a

y

-coordinate together in one subtraction.

Correction: Find the change in

x

using only

x

-coordinates, and the change in

y

using only

y

-coordinates.

Switching the point order in only one subtraction.

Correction: Subtract in the same order for both coordinates. For example, use second point minus first point for both.

Putting the slope in place of

b

in

y=mx+b

.

Correction: Remember that

m

is the slope. Find

b

by substituting a point into the equation.

Calling the slope of a vertical line zero.

Correction: A vertical line has zero run, so its slope is undefined. A horizontal line has zero rise and slope zero.

Lesson summary

Find slope by dividing the change in $y$ by the change in $x$ .
Keep the subtraction order consistent for both coordinates.
Use the slope and a known point to find the equation in the form $y=mx+b$ .
A vertical line has undefined slope; a horizontal line has slope zero.

Check your understanding

Question 1

What is the slope of the line through

(1,5)

and

(4,-1)

?

$-2$
$2$
$-\frac{1}{2}$
$6$

Show answer and explanation

-2

The change in

y

is

-1-5=-6

, and the change in

x

is

4-1=3

. The slope is

-6/3=-2

.

Question 2

A line has slope

3

and passes through

(2,7)

. What is its equation in the form

y=mx+b

?

$y=3x+1$
$y=3x+7$
$y=2x+1$
$y=3x-1$

Show answer and explanation

y=3x+1

Substitute the point into

y=mx+b

:

7=3(2)+b

, so

b=1

. Therefore the equation is

y=3x+1

.

Question 3

Which statement describes a vertical line?

Its slope is undefined because its run is zero.
Its slope is zero because its rise is zero.
Its slope is one because its rise and run are equal.
Its equation always has the form $y=mx+b$ with a number for $m$ .

Show answer and explanation

Its slope is undefined because its run is zero.

A vertical line has the same

x

-coordinate at every point. Its run is zero, and division by zero is not possible.

Key terms

Coordinate: A number that gives a point’s position on an axis.
Slope: The change in vertical position divided by the change in horizontal position.
Rise: The change in $y$ between two points.
Run: The change in $x$ between two points.
$y$ -intercept: The $y$ -coordinate where a line crosses the $y$ -axis.
Undefined slope: A slope that cannot be calculated because the change in $x$ is zero.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic G1. It is a study resource, not an official curriculum publication.

Official curriculum reference

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G1 · Find slope from two points and write a line equation

Ontario Grade 10 Mathematics

Use two points to describe a line’s change and write a rule for it.

What you will learn

1. Grade 9 bridge: coordinates and change

2. See rise and run, then use the slope rule

3. Use the slope and a point to write an equation

4. Special lines and independent practice

Reading rise and run from two points

Find the slope and equation

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MPM2D

About this lesson