Slope Calculator

Slope Formulas

1. Slope Formula (Two Points)

The slope of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

m = (y₂ - y₁) / (x₂ - x₁)

This formula calculates the rate of change of \(y\) with respect to \(x\), also known as "rise over run".

2. Slope-Intercept Form

The equation of a line can be written as:

y = mx + b

Where:

m is the slope of the line.
b is the y-intercept (where the line crosses the y-axis).

3. Point-Slope Form

If you know a point \((x₁, y₁)\) on the line and the slope \(m\), you can use the point-slope form:

y - y₁ = m(x - x₁)

This form is useful when you have a point and the slope, and you want to find the equation of the line.

4. Slope from an Angle

The slope of a line can also be related to the angle of inclination \(\theta\). If you know the angle, the slope is:

m = tan(θ)

Where \(\theta\) is the angle the line makes with the x-axis.

5. Perpendicular Slopes

If two lines are perpendicular, the product of their slopes is -1:

m₁ × m₂ = -1

Where m₁ and m₂ are the slopes of the two perpendicular lines.

6. Parallel Slopes

If two lines are parallel, their slopes are equal:

m₁ = m₂

Where m₁ and m₂ are the slopes of the two parallel lines.

7. Slope from Horizontal and Vertical Lines

For horizontal and vertical lines:

The slope of a horizontal line is \(m = 0\).
The slope of a vertical line is undefined.

8. Slope Between Two Parallel Lines

If two lines are parallel, they have the same slope. For example:

y₁ = m₁x + b₁

y₂ = m₁x + b₂

Both lines have the same slope \(m₁\).

9. Slope Formula for Distance and Slope

If you know the distance between two points and the x and y differences, you can find the slope using:

m = (Δy / Δx)

Where \(Δy\) is the difference in the y-coordinates and \(Δx\) is the difference in the x-coordinates.

10. General Equation of a Line

The general form of the equation of a line is:

Ax + By + C = 0

Where A, B, and C are constants. You can rearrange this to find the slope.

Advanced Slope Problems and Solutions

Problem 1: Find the slope of the line passing through points (3, 4) and (7, 8).

Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

\( m = \frac{8 - 4}{7 - 3} = \frac{4}{4} = 1 \)

The slope is \( m = 1 \).

Problem 2: Calculate the slope of a line with an incline angle of 30°.

Solution: Slope \( m = \tan(\theta) \):

\( m = \tan(30^\circ) \approx 0.577 \)

The slope is approximately \( m = 0.577 \).

Problem 3: Find the slope of a line parallel to the line passing through points (2, 5) and (8, 15).

Solution: First, calculate the slope of the original line:

\( m = \frac{15 - 5}{8 - 2} = \frac{10}{6} = \frac{5}{3} \)

The slope of any line parallel to this is also \( m = \frac{5}{3} \).

Problem 4: Find the slope of a line perpendicular to the line with slope \( m = \frac{4}{5} \).

Solution: The slope of a perpendicular line is the negative reciprocal of the original slope:

\( m_{\text{perpendicular}} = -\frac{5}{4} \)

The perpendicular slope is \( m = -\frac{5}{4} \).

Problem 5: Find the slope of the line with equation \( 3x - 4y = 12 \).

Solution: Rewrite in slope-intercept form \( y = mx + b \):

\( 3x - 4y = 12 \rightarrow y = \frac{3}{4}x - 3 \)

The slope is \( m = \frac{3}{4} \).

Problem 6: Determine the slope if a line rises 10 units for every 3 units it runs horizontally.

Solution: Slope \( m = \frac{\text{rise}}{\text{run}} \):

\( m = \frac{10}{3} \)

The slope is \( m = \frac{10}{3} \).

Problem 7: Calculate the slope for a line passing through (-2, 5) and (-2, -3).

Solution: This is a vertical line, so the slope is undefined.

Since \( x_1 = x_2 \), the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \) results in division by zero.

Problem 8: A line has slope \( m = -1.5 \). What is the angle of inclination of this line?

Solution: The angle \( \theta = \arctan(m) \):

\( \theta = \arctan(-1.5) \approx -56.31^\circ \)

Problem 9: Find the slope of the line perpendicular to \( y = -2x + 7 \).

Solution: The slope of the original line is \( m = -2 \).

The slope of a perpendicular line is \( m = \frac{1}{2} \).

Problem 10: For the line passing through points (1, 2) and (4, 8), is it increasing, decreasing, horizontal, or vertical?

Solution: Calculate the slope:

\( m = \frac{8 - 2}{4 - 1} = 2 \)

Since \( m > 0 \), the line is increasing.

Problem 11: Find the slope of the line passing through points (-4, -3) and (2, 5).

Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

\( m = \frac{5 - (-3)}{2 - (-4)} = \frac{8}{6} = \frac{4}{3} \)

The slope is \( m = \frac{4}{3} \).

Problem 12: What is the slope of a line parallel to \( y = 5x - 7 \)?

Solution: The slope of the given line is \( m = 5 \). Any parallel line will have the same slope.

The slope of the parallel line is \( m = 5 \).

Problem 13: Find the slope of a line perpendicular to \( y = 3x + 1 \).

Solution: The slope of the given line is \( m = 3 \). The slope of a perpendicular line is the negative reciprocal:

\( m_{\text{perpendicular}} = -\frac{1}{3} \)

The perpendicular slope is \( m = -\frac{1}{3} \).

Problem 14: Calculate the slope of the line with the equation \( 2x + 3y = 6 \).

Solution: Rewrite the equation in slope-intercept form \( y = mx + b \):

\( 2x + 3y = 6 \rightarrow y = -\frac{2}{3}x + 2 \)

The slope is \( m = -\frac{2}{3} \).

Problem 15: What is the slope of a line passing through the origin and the point (6, 8)?

Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) = (0, 0) \):

\( m = \frac{8 - 0}{6 - 0} = \frac{8}{6} = \frac{4}{3} \)

The slope is \( m = \frac{4}{3} \).

Problem 16: Find the slope of a line with equation \( y = -4x + 3 \).

Solution: The equation is already in slope-intercept form \( y = mx + b \), so the slope is \( m = -4 \).

The slope is \( m = -4 \).

Problem 17: What is the slope of a line passing through the points (5, -2) and (-3, 6)?

Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

\( m = \frac{6 - (-2)}{-3 - 5} = \frac{8}{-8} = -1 \)

The slope is \( m = -1 \).

Problem 18: Calculate the slope of a line with points (-1, 3) and (4, -2).

Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

\( m = \frac{-2 - 3}{4 - (-1)} = \frac{-5}{5} = -1 \)

The slope is \( m = -1 \).

Problem 19: What is the slope of a line passing through points (0, 0) and (10, 10)?

Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

\( m = \frac{10 - 0}{10 - 0} = \frac{10}{10} = 1 \)

The slope is \( m = 1 \).

Problem 20: Determine the slope of the line passing through the points (8, -4) and (6, -2).

Solution: Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

\( m = \frac{-2 - (-4)}{6 - 8} = \frac{2}{-2} = -1 \)

The slope is \( m = -1 \).