Algebra Fundamentals

Systems of Equations

• A system of equations is a set of two or more equations that must be satisfied simultaneously.
• Types of systems:
  • Independent systems: have a unique solution.
  • Dependent systems: have infinitely many solutions.
  • Inconsistent systems: have no solution.
• Methods for solving systems:
  • Substitution method: solving one equation for a variable and substituting into the other equation.
  • Elimination method: adding or subtracting equations to eliminate a variable.

Real Number Properties

• Properties of real numbers:
  • Commutative property of addition: a + b = b + a
  • Associative property of addition: (a + b) + c = a + (b + c)
  • Distributive property: a(b + c) = ab + ac
  • Additive identity: a + 0 = a
  • Additive inverse: a + (-a) = 0
  • Multiplicative identity: a × 1 = a
  • Multiplicative inverse: a × (1/a) = 1 (for non-zero a)

Linear Equations

• A linear equation is an equation in which the highest power of the variable(s) is 1.
• Standard form: ax + by = c, where a, b, and c are constants.
• Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
• Point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
• Solving linear equations:
  • Addition and subtraction
  • Multiplication and division
  • Graphing

Quadratic Equations

• A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.
• Types of quadratic equations:
  • Factored form: (x - r)(x - s) = 0, where r and s are roots.
  • Vertex form: a(x - h)^2 + k = 0, where (h, k) is the vertex.
• Solving quadratic equations:
  • Factoring
  • Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
  • Graphing