G6 Identify slope-intercept form and horizontal or vertical lines

Ontario Grade 10 Mathematics

Modelling Linear Relations

Identify line equations by their patterns and the coordinates they keep fixed

A line can be shown on a graph or described by an equation. An ordered pair gives a point’s position: the first number is its horizontal position, and the second is its vertical position. In this lesson, you will use equations to recognize slope-intercept form, horizontal lines, and vertical lines. These categories can overlap. For example, a horizontal line also fits slope-intercept form when its slope is zero.

What you will learn

Recognize an equation written in slope-intercept form.
Identify the slope and y-intercept in that form.
Recognize horizontal and vertical lines from their equations and graphs.
Explain why a vertical line is not written in slope-intercept form.
Recognize when an equation belongs to more than one category.

1. Bridge from coordinates and straight lines

A coordinate plane has a horizontal axis called the x-axis and a vertical axis called the y-axis. The point

(2,5)

is 2 units right and 5 units up from the origin, where the axes meet. The x-coordinate comes first; the y-coordinate comes second.

A line equation describes a relationship between the coordinates of points on the line. In the equation

y=4

, for example, the y-coordinate is always 4. The x-coordinate can change. This means the line stays at the same height as it goes across.

A line’s slope describes how much it rises or falls as you move horizontally. A horizontal line does not rise or fall, so its slope is 0. You will use this idea to identify a special case of slope-intercept form.

An ordered pair is written as $(x,y)$ : horizontal coordinate first, vertical coordinate second.
The x-axis is horizontal, and the y-axis is vertical.
An equation for a line tells which points lie on that line.

2. Recognize slope-intercept form

Slope-intercept form is a common way to write an equation for a line. Its pattern is

y=mx+b

. The letter

m

stands for the slope. The letter

b

stands for the y-intercept, the point where the line crosses the y-axis. At that point, the x-coordinate is 0.

To recognize this form, look for

y

alone on one side of the equals sign, and an expression involving

x

on the other side. The number multiplying

x

is the slope. The number added or subtracted is the y-intercept value. If no number is written before

x

, the coefficient is 1. For example, in

y=x-3

, the slope is 1 and the y-intercept value is

-3

.

Keep the sign of the constant term. In

y=2x-5

, the slope is 2 and the y-intercept value is

-5

. In

y=-3x+4

, the slope is

-3

and the y-intercept value is 4. The slope is attached to

x

; the y-intercept is the constant term.

A horizontal line is an important special case. The equation

y=6

can also be written as

y=0x+6

. It is both horizontal and in slope-intercept form. So do not assume that the line categories are always separate.

y=mx+b

Slope-intercept form follows the pattern $y=mx+b$ .
In that pattern, $m$ is the slope and $b$ is the y-intercept value.
A horizontal line can also be in slope-intercept form, with slope 0.

3. Identify horizontal and vertical lines

A horizontal line goes left and right at one fixed height. Each point on it has the same y-coordinate. Its equation has the form

y=c

, where

c

is a fixed number. For example,

y=3

describes a horizontal line. It is also slope-intercept form because it can be written as

y=0x+3

.

A vertical line goes up and down at one fixed horizontal position. Each point on it has the same x-coordinate. Its equation has the form

x=c

. For example,

x=-2

describes a vertical line because the x-coordinate is always

-2

.

A vertical line cannot be written in slope-intercept form. The pattern

y=mx+b

describes a line where

y

is given in terms of

x

. A vertical line keeps

x

fixed while

y

changes, so it does not fit that pattern.

You can use the same tests on a graph. A horizontal line stays level as you look across it. A vertical line stays in the same left-to-right position as you look up or down it. In an equation, check which coordinate stays fixed: fixed

y

means horizontal; fixed

x

means vertical.

y=c\qquad x=c

The pattern $y=c$ describes a horizontal line and is also slope-intercept form.
The pattern $x=c$ describes a vertical line and is not slope-intercept form.
More than one classification may apply to the same equation.

4. Practise identifying the patterns

For each equation, first check whether it matches

y=mx+b

. If it does, identify the slope and y-intercept value. Then check whether it is a special line. Classifications can overlap, so report every category that applies.

Try these independently. For

y=-2x+7

, name the form, slope, and y-intercept value. For

y=-4

, identify all applicable categories. For

x=5

, decide whether the line is horizontal or vertical and whether it is slope-intercept form.

As a check, ask which coordinate stays fixed. This works for equations and graphs. A fixed y-coordinate indicates a horizontal line; a fixed x-coordinate indicates a vertical line.

Check for slope-intercept form, then check for a special line.
Report overlapping categories when they apply.
Use the sign exactly as written when identifying a slope or y-intercept value.

Spot the line pattern

Equation pattern	What stays fixed?	Classification
$y=c$	The y-coordinate	Horizontal and slope-intercept form, with slope 0
$x=c$	The x-coordinate	Vertical; not slope-intercept form
$y=mx+b$	No coordinate must stay fixed	Slope-intercept form; horizontal when $m=0$

Worked example

Classify three line equations

For each equation, decide whether it is in slope-intercept form, horizontal, or vertical. More than one classification may apply. For any equation in slope-intercept form, name its slope and y-intercept value: (a)

y=3x-2

, (b)

y=5

, (c)

x=5

.

Inspect the first equation
The first equation has $y$ alone on the left and an expression with $x$ on the right. It matches slope-intercept form. The coefficient of $x$ is 3, and the constant term is $-2$ .
$y=3x-2$
Name its slope and y-intercept
In slope-intercept form, the coefficient of $x$ gives the slope, and the constant term gives the y-intercept value. The line crosses the y-axis at the point $(0,-2)$ .
$m=3,\quad b=-2$
Classify the second equation
The y-coordinate stays at 5 while the x-coordinate can change, so this is a horizontal line. It also fits slope-intercept form because the coefficient of $x$ is zero. Its slope is 0, and its y-intercept value is 5.
$y=5=0x+5$
Classify the third equation
The x-coordinate stays at 5 while the y-coordinate can change. This is a vertical line, so it is not in slope-intercept form.
$x=5$

Answer: (a) Slope-intercept form; slope 3; y-intercept value

-2

. (b) Horizontal and slope-intercept form; slope 0; y-intercept value 5. (c) Vertical; not slope-intercept form.

Check: For (b), every point has y-coordinate 5, confirming that the line is horizontal. Writing it as

y=0x+5

confirms slope-intercept form. For (c), every point has x-coordinate 5, confirming that the line is vertical.

Common mistakes and how to avoid them

Calling the constant term the slope in

y=3x-2

.

Correction: The slope is the number multiplying

x

, so it is 3. The y-intercept value is the constant term,

-2

.

Saying

y=5

is only a horizontal line.

Correction: It is horizontal because the y-coordinate is fixed. It is also slope-intercept form:

y=0x+5

.

Calling

x=5

a horizontal line because it contains a fixed number.

Correction: The equation fixes the x-coordinate, so the line is vertical. A fixed y-coordinate makes a line horizontal.

Treating a vertical line as slope-intercept form.

Correction: A vertical line fixes

x

while

y

changes. It does not fit the pattern

y=mx+b

.

Changing a negative y-intercept value to a positive one.

Correction: Keep the sign attached to the constant term. In

y=3x-2

, the y-intercept value is

-2

.

Lesson summary

Slope-intercept form is $y=mx+b$ , where $m$ is the slope and $b$ is the y-intercept value.
A horizontal line has a fixed y-coordinate and follows the pattern $y=c$ . It is also slope-intercept form with slope 0.
A vertical line has a fixed x-coordinate and follows the pattern $x=c$ . It is not slope-intercept form.
Some equations have more than one classification. Check each category rather than choosing only one.

Check your understanding

Question 1

In

y=-4x+6

, what is the slope?

$-4$
$6$
$4$
$-6$

Show answer and explanation

-4

The slope is the coefficient multiplying

x

, which is

-4

. The constant term 6 is the y-intercept value.

Question 2

Which equation describes a vertical line?

$y=-3$
$y=2x+1$
$x=-3$
$y=0x+3$

Show answer and explanation

x=-3

In

x=-3

, the x-coordinate stays fixed while the y-coordinate can change, so the line is vertical.

Question 3

What classifications apply to

y=-2

?

Horizontal only
Slope-intercept form only
Both horizontal and slope-intercept form
Vertical and slope-intercept form

Show answer and explanation

Both horizontal and slope-intercept form

The equation fixes the y-coordinate, so the line is horizontal. It can also be written as

y=0x-2

, so it is in slope-intercept form with slope 0.

Key terms

Slope: A number that describes how much a line rises or falls as you move horizontally.
Y-intercept: The point where a line crosses the y-axis; in $y=mx+b$ , the value $b$ is its y-coordinate.
Slope-intercept form: A way to write a line equation as $y=mx+b$ , where $m$ is slope and $b$ is the y-intercept value.
Horizontal line: A line that goes left and right at one fixed y-coordinate.
Vertical line: A line that goes up and down at one fixed x-coordinate.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic G6. It is a study resource, not an official curriculum publication.

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G6 · Identify slope-intercept form and horizontal or vertical lines

Ontario Grade 10 Mathematics

Identify line equations by their patterns and the coordinates they keep fixed

What you will learn

1. Bridge from coordinates and straight lines

2. Recognize slope-intercept form

3. Identify horizontal and vertical lines

4. Practise identifying the patterns

Spot the line pattern

Classify three line equations

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MFM2P

About this lesson