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Derive Linear Equations
Using Slope

Learn how slope stays constant between any two points on a line, and derive the equations \( y = mx \) and \( y = mx + b \) from similar triangles.

8th Grade Math

What is Slope?

Slope measures how steep a line is. It tells you how much \( y \) changes for every unit change in \( x \). We write it as a ratio:

A positive slope means the line goes up from left to right. A negative slope means it goes down. A slope of zero means the line is perfectly horizontal.

Slope on the Coordinate Plane

x y 1 2 3 4 5 1 2 3 4 (1, 1) (3, 2) run = 2 rise = 1

Similar Triangles Prove Constant Slope

Pick any two pairs of points on a non-vertical line and draw right triangles. The triangles are similar because they share the same angles: the line makes the same angle with the horizontal everywhere.

\(1\)
\( \frac{2}{3} \)
\(3\)
\(2\)
A B C D

Why the Triangles Are Similar

1
Both triangles are right triangles. The horizontal and vertical legs meet at \(90^\circ\).
2
They share the angle the line makes with the horizontal. This is the same everywhere on a straight line.
3
AA Similarity: Two equal angles guarantee the triangles are similar, so the ratio \( \frac{\text{rise}}{\text{run}} \) is constant.

Deriving \( y = mx \)

Start with a line through the origin \((0, 0)\). Pick any point \((x, y)\) on the line. The slope triangle from the origin to that point gives:

Multiply both sides by \( x \):

This equation describes every line through the origin with slope \( m \).

Deriving \( y = mx + b \)

Now consider a line that crosses the \( y \)-axis at \((0, b)\) instead of the origin. Pick any point \((x, y)\) on this line:

Multiply both sides by \( x \), then add \( b \):

Here \( b \) is the y-intercept: where the line meets the vertical axis.

\( y = mx \) vs \( y = mx + b \)

x y 1 2 -1 -2 1 2 -1 (0, 0) (0, 2)
\( y=x \)
\( y=x+2 \)
\( y = mx \)
Always passes through (0, 0). The \( y \)-intercept is zero.
\( y = mx + b \)
Passes through (0, b). The value \( b \) shifts the line up or down.

Slope vs Y-Intercept

Slope \( m \)
Controls the steepness and direction of the line. Changing \( m \) rotates the line.
Y-Intercept \( b \)
Controls where the line crosses the y-axis. Changing \( b \) shifts the line up or down without changing its steepness.

How to Find the Equation

1
Identify two points on the line, such as \((x_1, y_1)\) and \((x_2, y_2)\).
2
Calculate the slope: \( m = \dfrac{y_2 - y_1}{x_2 - x_1} \).
3
Find the y-intercept \( b \): Substitute one point and the slope into \( y = mx + b \), then solve for \( b \).
4
Write the equation: Plug \( m \) and \( b \) back into \( y = mx + b \).

Example 1: Line Through the Origin

A line passes through the origin and the point \((4, 6)\). Find its equation.

x y 1 2 3 4 5 1 2 3 4 5 6 (0, 0) (4, 6) run = 4 rise = 6

Example 2: Finding \( b \)

A line passes through \((2, 5)\) and \((6, 13)\). Find the equation in slope-intercept form.

1
Find slope: \( m = \dfrac{13 - 5}{6 - 2} = \dfrac{8}{4} = 2 \).
2
Find \( b \): Substitute \((2, 5)\) into \( y = 2x + b \): \( 5 = 2(2) + b \implies 5 = 4 + b \implies b = 1 \).
3
Write equation: \( y = 2x + 1 \).

Example 3: Negative Slope

A line passes through \((-1, 7)\) and \((3, -1)\). Find the equation.

1
Find slope: \( m = \dfrac{-1 - 7}{3 - (-1)} = \dfrac{-8}{4} = -2 \).
2
Find \( b \): Substitute \((3, -1)\): \( -1 = -2(3) + b \implies -1 = -6 + b \implies b = 5 \).
3
Equation: \( y = -2x + 5 \).

Common Mistakes

✕ Swapping Rise and Run
Writing \( m = \frac{x_2 - x_1}{y_2 - y_1} \) instead of \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Remember: rise over run.
✕ Inconsistent Point Order
Using \( y_2 - y_1 \) in the numerator but \( x_1 - x_2 \) in the denominator. Always subtract in the same order.
✕ Forgetting the Y-Intercept
Writing \( y = mx \) when the line does not pass through the origin. Always check if \( b = 0 \) before dropping it.

Summary of Key Formulas

Concept Formula When to Use
Slope \( m = \dfrac{y_2 - y_1}{x_2 - x_1} \) Given two points
Through Origin \( y = mx \) Line passes through \((0, 0)\)
Slope-Intercept \( y = mx + b \) Any non-vertical line
Similar Triangles \( \dfrac{\text{rise}_1}{\text{run}_1} = \dfrac{\text{rise}_2}{\text{run}_2} \) Proving slope is constant

Question 1 of 5

What is the slope of the line passing through \((2, 3)\) and \((5, 9)\)?

Question 2 of 5

A line passes through the origin with slope \( \dfrac{3}{4} \). Which equation represents this line?

Question 3 of 5

Why do similar triangles prove that slope is constant on a line?

Question 4 of 5

A line passes through \((1, 4)\) and \((3, 10)\). What is the equation in slope-intercept form?

Question 5 of 5

The equation of a line is \( y = -\dfrac{1}{2}x + 6 \). What is the y-intercept and what does the slope tell you?