Linear Equations: Slope of a Line

Q: What is the slope-intercept form of a linear equation?

The slope-intercept form is $y = ax + b$, where **$a$ is the slope** and **$b$ is the y-intercept**.

Q: How do we express $y$ as a function of $x$?

When we rewrite a linear equation to isolate $y$ in the form $y = ax + b$, the value of $y$ is determined entirely by $x$. This allows us to use function notation and write $y = f(x)$.

Understand slope as steps, calculate the slope between two points, and find the equation of a line.

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Slope-Intercept Form & Function Notation

The slope (represented by $a$ in the equation

y = ax + b

) describes the steepness and direction of a line.

Slope Steps Visualizer

For every step right (+1), we go up by 1.5.

Understanding Slope as Steps

Think of the slope as taking steps on the graph:
- Every time we move $1$ unit to the right (increase

x

1

), we go up or down by $a$ units (change

y

a

).
- If

a

is positive (e.g.

a = 2

), we go up by $2$ for every step right.
- If

a

is negative (e.g.

a = -1

), we move down by 1 for every step right.

The Slope Formula & Delta ( $\Delta$ )

To find the slope between two points

(x_1, y_1)

and

(x_2, y_2)

, we use the formula:

Slope of a Line

a = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

Calculated from two points

(x_1, y_1)

and

(x_2, y_2)

on the line.

What is Delta ( $\Delta$ )?
The Greek letter $\Delta$ (Delta) represents the change or difference between two values:
-

\Delta y = y_2 - y_1

is the change in the vertical direction (often called the rise).
-

\Delta x = x_2 - x_1

is the change in the horizontal direction (often called the run).

This is why the slope is often described as "rise over run" (

\frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}

).

⚠️ Crucial Order Note:
It does not matter which point you choose as the first point

(x_1, y_1)

and which as the second point

(x_2, y_2)

—the calculated slope will be exactly the same. However, you must maintain the same subtraction order in both the numerator and the denominator! If you subtract the

y

-coordinates as

y_2 - y_1

, you must subtract the

x

-coordinates as

x_2 - x_1

. Mixing the order (like

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

) is incorrect and will result in the wrong sign.

Slope Between Two Points

Slope Calculation:

a = \frac{\Delta y}{\Delta x} = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2

Finding the y-intercept $b$ and the Equation of a Line

Once we have the slope

a

, we can find the y-intercept

b

and write the full equation using a point

(x_1, y_1)

on the line. Since the point lies on the line, its coordinates must satisfy the equation

y = ax + b

.

Example:
Suppose we know the slope is

a = 3

, and the line passes through the point

(2, 7)

Start with the equation form: $y = ax + b$
Since $a = 3$ , this gives: $y = 3x + b$
Substitute the point $(x_1, y_1) = (2, 7)$ (where $x_1 = 2$ and $y_1 = 7$ ):
$y_1 = 3x_1 + b \implies 7 = 3 \cdot 2 + b$
Solve for $b$ :
$7 = 6 + b \implies b = 1$
The full equation is: $\boxed{y = 3x + 1}$

Equation of a Line from Two Points

If we are given only two points, we can determine the entire equation by combining the two steps:
1. Calculate the slope

a

from the two points using the slope formula.
2. Use the calculated slope

a

and one of the points to find the y-intercept

b

(exactly as shown in the example above).

Parallel Lines

Two lines are parallel if they lie in the same plane and never intersect, no matter how far they are extended. They rise or fall by the same amount for every unit moved horizontally, which means they have the same slope (

a_1 = a_2

), but different y-intercepts (

b_1 \neq b_2

).

Example:
- Line 1:

y = 2x + 3

(slope

a = 2

, y-intercept

b = 3

)
- Line 2:

y = 2x - 5

(slope

a = 2

, y-intercept

b = -5

)
Both lines have a slope of

2

, so they are parallel. Because they cross the y-axis at different heights (

3

and

-5

), they are distinct, non-overlapping lines that will never touch.

Parallel Lines Visualizer

Select Shared Slope (a):

Blue Intercept (b₁):

Red Intercept (b₂):

y = 1.5x + 1

y = 1.5x - 2

Property of Parallel Lines

The graphs of two linear functions are parallel if and only if they have the same slope:

a_1 = a_2

(and different y-intercepts:

b_1 \neq b_2

Note: "If and only if" means the rule works in both directions: if the lines are parallel, their slopes must be equal; and if their slopes are equal, the lines must be parallel.

Learning Topics

Frequently Asked Questions

What is the slope-intercept form of a linear equation?

The slope-intercept form is

y = ax + b

, where $a$ is the slope and $b$ is the y-intercept.

How do we express

y

as a function of

x

When we rewrite a linear equation to isolate

y

in the form

y = ax + b

, the value of

y

is determined entirely by

x

. This allows us to use function notation and write

y = f(x)