Textbook

1. Word Knowledge

2. Mathematics Knowledge

2.1 Foundational math

2.2 Pre-algebra

2.3 Algebra I

2.4 Geometry basics

2.5 Algebra II

2.6 Advanced geometry

2.6.1 Equations of lines and slope

2.6.2 Right triangles and similar triangles

2.6.3 Distance and circles

2.6.4 Radians and measurement applications

2.7 Math strategies

3. Paragraph Comprehension

4. Arithmetic Reasoning

5. General Science

6. Mechanical Comprehension

7. Electronics Information

8. Shop Information

9. Auto Information

10. Assembling Objects

Wrapping up

2.6.1 Equations of lines and slope

Achievable ASVAB

2. Mathematics Knowledge

2.6. Advanced geometry

Our ASVAB course is currently in development and is a work-in-progress.

Equations of lines and slope

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Finding equations of lines

An equation of a line describes all the points that lie on that line. One of the most common ways to write this equation is in slope-intercept form, which shows both the slope and where the line crosses the $y$ -axis.

$y = m x + b$

This equation describes all coordinate points $(x, y)$ that lie on the line. Here’s what each part means:

$x$ and $y$ form any coordinate point on the line
$m$ is the slope of the line (from the previous section, $s l o p e = \frac{r i se}{r u n}$ , which tells you the steepness and direction of the line)
$b$ is the $y$ -intercept, where the line crosses the $y$ -axis

So if we have the line $y = 2 x - 4$ , then:

The slope is $2$ (or $\frac{2}{1}$ )
The $y$ -intercept is $y = - 4$

The line y = 2x − 4 on a coordinate plane, rising from left to right, crossing the y-axis at (0, −4) and the x-axis at (2, 0). — y=2x-4

Slope

The slope of a line tells you how steep the line is and in which direction it moves as you go from left to right. You can find the slope using two points on the line. You can also find the equation of a line by starting with the slope formula (using two points on the line):

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Let’s solve this example step-by-step!

Example:

Suppose you want the equation of the line that passes through the points $(- 2, 7)$ and $(4, 1)$ .

Step 1: Find the slope. Start with the slope formula and substitute the two points.

$m m m m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = \frac{1 - 7}{4 - ( - 2 )} = \frac{- 6}{6} = - 1$

Step 2: Substitute the slope into $y = m x + b$ .

$y = - 1 x + b$

Step 3: Use a point to solve for $b$ . Any point on the line will satisfy the equation. Using $(- 2, 7)$ :

$775 = - 1 (- 2) + b = 2 + b = b$

So the final equation of the line is:

$y = - x + 5$

Knowledge check

What is the equation for the line that passes through points $(- 1, - 1)$ and $(3, 15)$ ?

A. $y = 2 x - 6$
B. $y = - 2 x + 4$
C. $y = - 4 x - 2$
D. $y = 4 x + 3$

(spoiler)

D. $y = 4 x + 3$

Slope-intercept form

Equation: $y = m x + b$
$m$ = slope (steepness/direction), $b$ = $y$ -intercept (where line crosses $y$ -axis)
$x, y$ represent any point on the line
Example: $y = 2 x - 4$ has slope $2$ , y-intercept $- 4$

Finding slope from two points

Formula: $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Used to calculate steepness/direction before writing full equation

Deriving a line’s equation from two points

Steps:
1. Calculate slope $m$ using slope formula
2. Plug $m$ into $y = m x + b$
3. Substitute one known point’s coordinates to solve for $b$
Final equation combines solved $m$ and $b$ into $y = m x + b$

Next | 2.6.2 Right triangles and similar triangles

Equations of lines and slope

Finding equations of lines

$y = m x + b$

This equation describes all coordinate points $(x, y)$ that lie on the line. Here’s what each part means:

$x$ and $y$ form any coordinate point on the line
$m$ is the slope of the line (from the previous section, $s l o p e = \frac{r i se}{r u n}$ , which tells you the steepness and direction of the line)
$b$ is the $y$ -intercept, where the line crosses the $y$ -axis

So if we have the line $y = 2 x - 4$ , then:

The slope is $2$ (or $\frac{2}{1}$ )
The $y$ -intercept is $y = - 4$

Slope

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Let’s solve this example step-by-step!

Example:

Suppose you want the equation of the line that passes through the points $(- 2, 7)$ and $(4, 1)$ .

Step 1: Find the slope. Start with the slope formula and substitute the two points.

$m m m m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = \frac{1 - 7}{4 - ( - 2 )} = \frac{- 6}{6} = - 1$

Step 2: Substitute the slope into $y = m x + b$ .

$y = - 1 x + b$

Step 3: Use a point to solve for $b$ . Any point on the line will satisfy the equation. Using $(- 2, 7)$ :

$775 = - 1 (- 2) + b = 2 + b = b$

So the final equation of the line is:

$y = - x + 5$

Knowledge check

What is the equation for the line that passes through points $(- 1, - 1)$ and $(3, 15)$ ?

A. $y = 2 x - 6$
B. $y = - 2 x + 4$
C. $y = - 4 x - 2$
D. $y = 4 x + 3$

(spoiler)

D. $y = 4 x + 3$

Achievable ASVAB

2. Mathematics Knowledge

2.6. Advanced geometry

Our ASVAB course is currently in development and is a work-in-progress.

Equations of lines and slope

3 min read

Font

Discuss

Feedback

Finding equations of lines

$y = m x + b$

This equation describes all coordinate points $(x, y)$ that lie on the line. Here’s what each part means:

$x$ and $y$ form any coordinate point on the line
$m$ is the slope of the line (from the previous section, $s l o p e = \frac{r i se}{r u n}$ , which tells you the steepness and direction of the line)
$b$ is the $y$ -intercept, where the line crosses the $y$ -axis

So if we have the line $y = 2 x - 4$ , then:

The slope is $2$ (or $\frac{2}{1}$ )
The $y$ -intercept is $y = - 4$

Slope

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Let’s solve this example step-by-step!

Example:

Suppose you want the equation of the line that passes through the points $(- 2, 7)$ and $(4, 1)$ .

Step 1: Find the slope. Start with the slope formula and substitute the two points.

$m m m m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = \frac{1 - 7}{4 - ( - 2 )} = \frac{- 6}{6} = - 1$

Step 2: Substitute the slope into $y = m x + b$ .

$y = - 1 x + b$

Step 3: Use a point to solve for $b$ . Any point on the line will satisfy the equation. Using $(- 2, 7)$ :

$775 = - 1 (- 2) + b = 2 + b = b$

So the final equation of the line is:

$y = - x + 5$

Knowledge check

What is the equation for the line that passes through points $(- 1, - 1)$ and $(3, 15)$ ?

A. $y = 2 x - 6$
B. $y = - 2 x + 4$
C. $y = - 4 x - 2$
D. $y = 4 x + 3$

(spoiler)

D. $y = 4 x + 3$

Slope-intercept form

Equation: $y = m x + b$
$m$ = slope (steepness/direction), $b$ = $y$ -intercept (where line crosses $y$ -axis)
$x, y$ represent any point on the line
Example: $y = 2 x - 4$ has slope $2$ , y-intercept $- 4$

Finding slope from two points

Formula: $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Used to calculate steepness/direction before writing full equation

Deriving a line’s equation from two points

Steps:
1. Calculate slope $m$ using slope formula
2. Plug $m$ into $y = m x + b$
3. Substitute one known point’s coordinates to solve for $b$
Final equation combines solved $m$ and $b$ into $y = m x + b$

Next | 2.6.2 Right triangles and similar triangles

Equations of lines and slope

Finding equations of lines

$y = m x + b$

This equation describes all coordinate points $(x, y)$ that lie on the line. Here’s what each part means:

$x$ and $y$ form any coordinate point on the line
$m$ is the slope of the line (from the previous section, $s l o p e = \frac{r i se}{r u n}$ , which tells you the steepness and direction of the line)
$b$ is the $y$ -intercept, where the line crosses the $y$ -axis

So if we have the line $y = 2 x - 4$ , then:

The slope is $2$ (or $\frac{2}{1}$ )
The $y$ -intercept is $y = - 4$

Slope

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Let’s solve this example step-by-step!

Example:

Suppose you want the equation of the line that passes through the points $(- 2, 7)$ and $(4, 1)$ .

Step 1: Find the slope. Start with the slope formula and substitute the two points.

$m m m m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = \frac{1 - 7}{4 - ( - 2 )} = \frac{- 6}{6} = - 1$

Step 2: Substitute the slope into $y = m x + b$ .

$y = - 1 x + b$

Step 3: Use a point to solve for $b$ . Any point on the line will satisfy the equation. Using $(- 2, 7)$ :

$775 = - 1 (- 2) + b = 2 + b = b$

So the final equation of the line is:

$y = - x + 5$

Knowledge check

What is the equation for the line that passes through points $(- 1, - 1)$ and $(3, 15)$ ?

A. $y = 2 x - 6$
B. $y = - 2 x + 4$
C. $y = - 4 x - 2$
D. $y = 4 x + 3$

(spoiler)

D. $y = 4 x + 3$