ASVAB Study Guide
Algebra · Lesson 6 of 13

Calculating Slope

Key rules and formulas, worked examples and 12 practice problems with step-by-step solutions.

1. Slope Formula

Finding slope from two points

$$m = \frac{(y_2 - y_1)}{(x_2 - x_1)}$$

Subtract the y-coordinates to find the ""Rise"" and the x-coordinates to find the ""Run.""

Example

Find the slope of the line passing through (2, 3) and (6, 11).

Show solution

Step 1: Rise = $$y_2 - y_1 = 11 - 3 = 8$$.

Step 2: Run $$= x_2 - x_1 = 6 - 2 = 4$$.

Step 3: Slope $$m = \frac{8}{4} = 2$$.

Result: $$m = 2$$

Practice problems

  1. Problem 1

    Find the slope of the line passing through (1, 2) and (4, 8).

    Show solution

    Answer: $$m = 2$$

    • Step 1: Identify points: $$(x_1, y_1) = (1, 2)$$ and $$(x_2, y_2) = (4, 8)$$.
    • Step 2: Calculate Rise $$= y_2 - y_1 = 8 - 2 = 6$$.
    • Step 3: Calculate Run $$= x_2 - x_1 = 4 - 1 = 3$$.
    • Step 4: Compute slope $$m = \frac{\text{Rise}}{\text{Run}} = \frac{6}{3} = 2$$.

    => Correct Option: B ($$m = 2$$)

  2. Problem 2

    Find the slope of the line passing through $$(-2, 5)$$ and $$(3, -5)$$

    Show solution

    Answer: $$m = -2$$

    • Step 1: Identify points: $$(x_1, y_1) = (-2, 5)$$ and $$(x_2, y_2) = (3, -5)$$.
    • Step 2: Calculate Rise $$= y_2 - y_1 = -5 - 5 = -10$$.
    • Step 3: Calculate Run $$= x_2 - x_1 = 3 - (-2) = 5$$.
    • Step 4: Compute slope $$m = \frac{-10}{5} = -2$$.

    => Correct Option: B ($$m = -2$$)

  3. Problem 3

    Find the slope of the line passing through $$(-4, -7)$$ and (2, 5).

    Show solution

    Answer: $$m = 2$$

    • Step 1: Identify points: $$(x_1, y_1) = (-4, -7)$$ and $$(x_2, y_2) = (2, 5)$$.
    • Step 2: Calculate Rise $$= y_2 - y_1 = 5 - (-7) = 5 + 7 = 12$$.
    • Step 3: Calculate Run $$= x_2 - x_1 = 2 - (-4) = 2 + 4 = 6$$.
    • Step 4: Compute slope $$m = \frac{12}{6} = 2$$.

    => Correct Option: B ($$m = 2$$)

Remember

Subtract the y-coordinates to find the 'Rise' (change in y). Subtract the x-coordinates to find the 'Run' (change in x).

Formula: $$m = \frac{\text{Rise}}{\text{Run}} = \frac{(y_2 - y_1)}{(x_2 - x_1)}$$. Ensure consistent point ordering!

2. From Equation

Finding slope from Slope-Intercept Form

$$y = mx + b$$

When the equation is in this form, the coefficient of x is the slope.

Example

Find the slope of the line $$y = -3x + 7$$

Show solution

The equation is in $$y = mx + b$$ form.

Here, $$m = -3$$ and $$b = 7$$.

Result: Slope $$m = -3$$

Practice problems

  1. Problem 4

    Find the slope of the line $$y = 4x - 5$$

    Show solution

    Answer: $$m = 4$$

    • Step 1: The equation $$y = 4x - 5$$ is already in slope-intercept form $$y = mx + b$$.
    • Step 2: Match the coefficient of x with m: $$m = 4$$.
    • Step 3: Therefore, the slope of the line is $$m = 4$$.

    => Correct Option: A ($$m = 4$$)

  2. Problem 5

    Find the slope of the line $$y = \frac{-2}{3} x + 1$$

    Show solution

    Answer: $$m = \frac{-2}{3}$$

    • Step 1: Compare $$y = \frac{-2}{3} x + 1$$ with $$y = mx + b$$.
    • Step 2: The coefficient in front of x is $$\frac{-2}{3}$$.
    • Step 3: Therefore, the slope is $$m = \frac{-2}{3}$$.

    => Correct Option: B ($$m = \frac{-2}{3}$$)

  3. Problem 6

    Find the slope of the line $$3x + 2y = 12$$

    Show solution

    Answer: $$m = \frac{-3}{2}$$

    • Step 1: Convert Standard Form $$3x + 2y = 12$$ to Slope-Intercept Form.
    • Step 2: Subtract $$3x$$ from both sides: $$2y = -3x + 12$$.
    • Step 3: Divide all terms by 2: $$y = (\frac{-3}{2})x + 6$$.
    • Step 4: Identify coefficient of x: $$m = \frac{-3}{2}$$.

    => Correct Option: A ($$m = \frac{-3}{2}$$)

Remember

When the equation is in slope-intercept form ($$y = mx + b$$), the coefficient of x ('m') is the slope.

If in standard form (Ax + By = C), rearrange to $$y = mx + b$$ or use $$m = -\frac{A}{B}$$

3. From a Graph

Counting units on a coordinate plane

$$m = \text{Rise} / \text{Run}$$

Pick two points on the line. Count how many units you go up/down and left/right.

Example

A line goes up 6 units (Rise = +6) and right 2 units (Run = +2) between two grid points. Find the slope.

Show solution

Rise = 6, Run = 2.

$$m = \frac{\text{Rise}}{\text{Run}} = \frac{6}{2} = 3$$.

Result: Slope $$m = 3$$

Practice problems

  1. Problem 7

    A line goes up 3 units and right 1 unit between two points. Find the slope.

    Show solution

    Answer: $$m = 3$$

    • Step 1: Vertical change (Rise) = +3 (going up).
    • Step 2: Horizontal change (Run) = +1 (going right).
    • Step 3: Apply formula $$m = \frac{\text{Rise}}{\text{Run}} = \frac{3}{1} = 3$$.

    => Correct Option: A ($$m = 3$$)

  2. Problem 8

    A line goes down 4 units and right 2 units between two points. Find the slope.

    Show solution

    Answer: $$m = -2$$

    • Step 1: Vertical change (Rise) = -4 (going down).
    • Step 2: Horizontal change (Run) = +2 (going right).
    • Step 3: Apply formula $$m = \frac{\text{Rise}}{\text{Run}} = \frac{-4}{2} = -2$$.

    => Correct Option: B ($$m = -2$$)

  3. Problem 9

    A line goes down 5 units and left 2 units between two points. Find the slope.

    Show solution

    Answer: $$m = \frac{5}{2}$$

    • Step 1: Vertical change (Rise) = -5 (going down).
    • Step 2: Horizontal change (Run) = -2 (going left).
    • Step 3: Apply formula $$m = \frac{\text{Rise}}{\text{Run}} = \frac{(-5)}{(-2)} = \frac{5}{2}$$.

    => Correct Option: B ($$m = \frac{5}{2}$$)

Remember

Pick two distinct grid points on the line. Count vertical change ('Rise': up positive, down negative) and horizontal change ('Run': right positive, left negative).

$$m = \frac{\text{Rise}}{\text{Run}}$$.

4. Special Slopes

Horizontal and Vertical lines

Horiz: $$m = 0$$; Vert: $$m = \text{Undefined}$$

Horizontal lines have no ""Rise."" Vertical lines have no ""Run"" (cannot divide by zero).

Example

Find the slope of the line $$y = 4$$ and the line $$x = -2$$

Show solution

For $$y = 4$$: Horizontal line, Rise $$= 0 \Rightarrow m = 0$$.

For $$x = -2$$: Vertical line, Run $$= 0 \Rightarrow m$$ is Undefined.

Practice problems

  1. Problem 10

    What is the slope of the horizontal line $$y = -7$$?

    Show solution

    Answer: $$m = 0$$

    • Step 1: Recognize that $$y = -7$$ represents a horizontal line.
    • Step 2: On a horizontal line, y-values never change (Rise = 0).
    • Step 3: Slope $$m = \frac{\text{Rise}}{\text{Run}} = \frac{0}{\text{Run}} = 0$$.

    => Correct Option: A ($$m = 0$$)

  2. Problem 11

    What is the slope of the vertical line $$x = 5$$?

    Show solution

    Answer: Undefined

    • Step 1: Recognize that $$x = 5$$ represents a vertical line.
    • Step 2: On a vertical line, x-values never change (Run = 0).
    • Step 3: Slope $$m = \frac{\text{Rise}}{0}$$, which is Undefined (division by zero is impossible).

    => Correct Option: B (Undefined).

  3. Problem 12

    Find the slope of the line passing through (4, 9) and $$(4, -2)$$

    Show solution

    Answer: Undefined

    • Step 1: Notice x-coordinates are identical: $$x_1 = 4$$, $$x_2 = 4$$.
    • Step 2: Calculate Run $$= x_2 - x_1 = 4 - 4 = 0$$.
    • Step 3: Calculate Rise $$= y_2 - y_1 = -2 - 9 = -11$$.
    • Step 4: $$m = \frac{-11}{0}$$ => Undefined slope (Vertical Line).

    => Correct Option: B (Undefined).

Remember

1. Horizontal lines $$(y = k)$$ have no vertical change (Rise = 0), so $$m = \frac{0}{\text{Run}} = 0$$.

2. Vertical lines $$(x = k)$$ have no horizontal change (Run = 0), so $$m = \frac{\text{Rise}}{0} = \text{Undefined}$$ (cannot divide by zero).