Linear Equations - Theory & Concepts

Linear Equations

A linear equation represents a straight line on the Cartesian coordinate plane. It is the foundation for modeling constant rates of change and solving problems involving speed, cost, and mixtures.

Forms of Linear Equations

Depending on the information available, linear equations can be expressed in different forms.

Slope-Intercept Form: $y = mx + b$ . Identifies slope $m$ and y-intercept $b$ .
Point-Slope Form: $y - y_1 = m(x - x_1)$ . Used when a point $(x_1, y_1)$ and slope $m$ are known.
Standard Form: $Ax + By = C$ . Useful for finding intercepts where $A \ge 0$ .

Note

Use the interactive converter below to explore how linear equations switch between Point-Slope form, Slope-Intercept form, and Standard form. Drag the coordinates or adjust the sliders to see the transformations in real-time.

Linear Equation Forms Converter

Point-Slope Coordinates

Point 1:

P_1(x_1, y_1)

x_1

-2

y_1

-3

Point 2:

P_2(x_2, y_2)

x_2

y_2

Active Conversions

Point-Slope

y - (-3) = 1.00(x - (-2))

Slope-Intercept

y = 1.00x - 1.00

Standard Form (Ax + By = C)

(5.0)x - (5.0)y = 5.0

Rise / Run5 / 5

Slope (

m

)1.00

y-Intercept (

b

)-1.00

Distance and Midpoint

Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a Cartesian plane, we can find the distance between them and their exact center.

Distance Formula

Calculates the straight-line distance between two points (derived from the Pythagorean theorem).

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Variables

Symbol	Description	Unit
$d$	The distance between the two points	units
$(x_{1}, y_{1})$	Coordinates of the first point	-
$(x_{2}, y_{2})$	Coordinates of the second point	-

Midpoint Formula

Calculates the exact center coordinate between two points.

M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Variables

Symbol	Description	Unit
$M$	The midpoint coordinates (x_m, y_m)	-
$(x_{1}, y_{1})$	Coordinates of the first point	-
$(x_{2}, y_{2})$	Coordinates of the second point	-

Understanding Slope

The slope $m$ represents the "Rise over Run", indicating the steepness and direction of the line.

Positive Slope: Line rises from left to right.
Negative Slope: Line falls from left to right.
Zero Slope: Horizontal line ( $y = b$ ).
Undefined Slope: Vertical line ( $x = a$ ).

Slope Formula

Calculates the slope (steepness) of a line given two points.

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

Variables

Symbol	Description	Unit
$m$	The slope of the line	-
$\Delta y$	The change in the y-coordinates (rise)	-
$\Delta x$	The change in the x-coordinates (run)	-
$(x_{1}, y_{1})$	Coordinates of the first point	-
$(x_{2}, y_{2})$	Coordinates of the second point	-

Interactive Visualizer

Note

Experiment with the slope $m$ and y-intercept $b$ below. Observe how changing $m$ rotates the line and changing $b$ shifts it vertically.

Linear Equation Explorer

Slope m1

Y-intercept b0

Current form

y = x

Converted meaning

y = x

Standard form converts by solving $A x + B y = C$ for $y$ when $B \\neq 0$ . If $B = 0$ , the graph is vertical and the slope is undefined.

State

Line with slope 1 and y-intercept 0

Slope

Intercept cue

y-intercept 0

Parallel and Perpendicular Lines

Slopes reveal the geometric relationship between two lines.

Parallel Lines: Have equal slopes ( $m_1 = m_2$ ). They never intersect.
Perpendicular Lines: Have slopes that are negative reciprocals ( $m_1 \cdot m_2 = -1$ ).

Interactive Simulation

Note

Below is an interactive calculator to explore the properties of $y = mx + b$ . You can change the slope $m$ and y-intercept $b$ to see how the line changes in real-time.

Slope-Intercept Explorer ($y = mx + b$)

y = 1.5 x + 1

Slope m (steepness & direction)1.5

Y-Intercept b (vertical shift)1

Interactive Insights

Slope ($m = 1.5$): Line rises from left to right.

Y-Intercept ($b = 1$): The line intersects the vertical y-axis exactly at point $(0, 1)$.

\text{Slope} = \frac{\text{Rise}}{\text{Run}} = \frac{3.0}{2} = 1.5

Y-Intercept (0, b)

Run (Horizontal)

Rise (Vertical)

Key Takeaways

Slope-Intercept Form: $y=mx+b$ is the most useful for graphing and understanding the "starting value" $b$ .
Special Slopes: Horizontal lines have slope $0$ ( $y=b$ ); Vertical lines have an undefined slope ( $x=a$ ).
Parallel vs Perpendicular: Parallel lines have the same slope; Perpendicular lines have negative reciprocal slopes.
Golden Rule: When solving linear inequalities, flip the inequality sign if multiplying or dividing by a negative number.

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