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:

Common Equation Forms

  • Slope-Intercept Form: y=mx+by = mx + b. Identifies slope (mm) and y-intercept (bb).
  • Point-Slope Form: y−y1=m(x−x1)y - y_1 = m(x - x_1). Used when a point (x1,y1)(x_1, y_1) and slope (mm) are known.
  • Standard Form: Ax+By=CAx + By = C. Useful for finding intercepts (A≥0A \ge 0).

Distance and Midpoint

Given two points (x1,y1)(x_1, y_1) and (x2,y2)(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=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Variables

SymbolDescriptionUnit
ddThe distance between the two points-
(x1,y1)(x_1, y_1)Coordinates of the first point-
(x2,y2)(x_2, y_2)Coordinates of the second point-

Midpoint Formula

Calculates the exact center coordinate between two points.

M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Variables

SymbolDescriptionUnit
MMThe midpoint coordinates (x_m, y_m)-
(x1,y1)(x_1, y_1)Coordinates of the first point-
(x2,y2)(x_2, y_2)Coordinates of the second point-

Understanding Slope

The slope (mm) represents the "Rise over Run", indicating the steepness and direction of the line.

Slope Formula

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

m=ΔyΔx=y2−y1x2−x1m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

Variables

SymbolDescriptionUnit
mmThe slope of the line-
Δy\Delta yThe change in the y-coordinates (rise)-
Δx\Delta xThe change in the x-coordinates (run)-
(x1,y1)(x_1, y_1)Coordinates of the first point-
(x2,y2)(x_2, y_2)Coordinates of the second point-

Slope Types

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

Interactive Visualizer

Experiment with the slope (mm) and y-intercept (bb) below. Observe how changing mm rotates the line and changing bb shifts it vertically.

Linear Equation Explorer:y=mx+by = mx + b

Slope (m)1

Controls steepness and direction.

Y-Intercept (b)0

Where the line crosses the Y-axis.

y=xy = x
xy

Parallel and Perpendicular Lines

Slopes reveal the geometric relationship between two lines:

Relational Slope Rules

  • Parallel Lines: Have equal slopes (m1=m2m_1 = m_2). They never intersect.
  • Perpendicular Lines: Have slopes that are negative reciprocals (m1⋅m2=−1m_1 \cdot m_2 = -1).
Key Takeaways
  • Slope-Intercept Form (y=mx+by=mx+b): Most useful for graphing and understanding the "starting value" (bb).
  • Special Slopes: Horizontal lines have slope 0 (y=by=b); Vertical lines have undefined slope (x=ax=a).
  • Parallel vs Perpendicular: Parallel = same slope; Perpendicular = negative reciprocal slope.
  • Golden Rule: When solving linear inequalities, flip the sign if multiplying/dividing by a negative.
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