Linear Equations

Linear Equations

Forms of Linear Equations

Form Equation When to Use
Slope-intercept Graphing, identifying slope/intercept
Standard Finding intercepts, integer solutions
Point-slope When given a point and slope

Conversions

Standard → Slope-intercept:

Point-slope → Slope-intercept:

Slope

Slope Direction
Line rises left → right
Line falls left → right
Horizontal line ()
Undefined Vertical line ()

Parallel and Perpendicular

  • Parallel lines: (same slope, different intercept)
  • Perpendicular lines: (negative reciprocal slopes)

Solving Linear Equations (One Variable)

Method:

  1. Distribute and simplify each side
  2. Collect variable terms on one side, constants on the other
  3. Divide to isolate the variable

Number of solutions:

Result Solutions Meaning
Exactly one Unique solution
(e.g., ) None Contradiction — inconsistent equation
Infinitely many Identity — both sides are the same

Example — No solution:
No solution when (constants differ after variables cancel).

Finding an Equation

Given slope and a point :

Given two points:

  2. Plug and one point into point-slope form

Given x-intercept and y-intercept :

Interpreting in Context (SAT Favorite)

In :

  • (slope) = rate of change per unit of
  • (y-intercept) = initial value (value when )

"A plumber charges $50 plus $30 per hour"
→ slope is hourly rate, intercept is flat fee

SAT Tips

  1. Equations with fractions — multiply every term by the LCD first
  2. "What does the number represent?" — always slope (rate) or y-intercept (starting value)
  3. No solution / infinite solutions — compare coefficients: if variable coefficients are proportional but constants are not → no solution; if everything is proportional → infinite
  4. Creating equations from tables — pick two rows to find slope, then use point-slope
  5. Quick intercepts from standard form : x-intercept = , y-intercept =
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