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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