Algebra

Algebra Formula Reference

Linear Equations

FormFormulaUse
Standardax+b=cax + b = cSolving for xx: x=c−bax = \frac{c - b}{a}
Variables both sidesax+b=cx+dax + b = cx + dx=d−ba−cx = \frac{d - b}{a - c}

Solving Steps

  1. Distribute (remove parentheses)
  2. Combine like terms on each side
  3. Move variable terms to one side
  4. Move constants to the other side
  5. Divide by the coefficient

Slope

ConceptFormula
Slope from two pointsm=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
Rise over runm=riserunm = \frac{\text{rise}}{\text{run}}
Positive slopeLine goes up left to right
Negative slopeLine goes down left to right
Zero slopeHorizontal line (y=cy = c)
Undefined slopeVertical line (x=cx = c)

Graphing Linear Equations

FormFormulaKey Info
Slope-intercepty=mx+by = mx + bmm = slope, bb = y-intercept
Point-slopey−y1=m(x−x1)y - y_1 = m(x - x_1)Through (x1,y1)(x_1, y_1) with slope mm
StandardAx+By=CAx + By = Cx-intercept: (C/A,0)(C/A, 0), y-intercept: (0,C/B)(0, C/B)

Intercepts

  • x-intercept: Set y=0y = 0, solve for xx
  • y-intercept: Set x=0x = 0, solve for yy

Systems of Equations

Substitution Method

  1. Solve one equation for one variable
  2. Substitute into the other equation
  3. Solve, then back-substitute

Elimination Method

  1. Multiply equations to match coefficients
  2. Add/subtract to eliminate one variable
  3. Solve, then back-substitute

Solution Types

TypeMeaningGraph
One solutionLines intersectCross at one point
No solutionLines are parallelSame slope, different intercept
Infinite solutionsLines overlapSame equation

Inequalities

SymbolMeaningNumber Line
<<Less thanOpen circle, shade left
>>Greater thanOpen circle, shade right
≤\leqLess than or equalClosed circle, shade left
≥\geqGreater than or equalClosed circle, shade right

Key rule: When multiplying or dividing both sides by a negative number, flip the inequality sign.

Compound Inequalities

TypeMeaningExample
ANDBoth must be true2<x<72 < x < 7
ORAt least one must be truex<1x < 1 or x>5x > 5