Algebra 2 Unit 3 Study Guide

System of Linear Equations

• A system consists of two linear equations in two variables, typically represented as:
  • Equation 1: Ax + By = C
  • Equation 2: Dx + Ey = F

Solution of a System

• A solution for the system is an ordered pair (x, y) that satisfies both equations simultaneously.

Evaluating Solutions

• To verify if (1, 3) is a solution:
  • Substitute x = 1 and y = 3 into both equations:
  • For Equation 1: 5(1) - 3(3) = -4 (valid)
  • For Equation 2: 1 + 2(3) = 7 (valid)

Graphical Representation

• The graph of a system can visually indicate the solution's approximation, which should be confirmed algebraically for precision in coordinates.

Solving Linear Systems

• To solve a system, isolate y in one equation and substitute into the other:
  • For example:
    • Start with -2x + y = 8 and 3x + y = -2.
    • Rearrange to y = 2x + 8 and substitute into the second equation.
    • Solve to find x = -2, then substitute back to find y = 4.
    • Final solution: (-2, 4).

Cattle and Sheep Grazing Model

• Between 1970 and 1990, models for the number (in thousands) of cattle and sheep on national forest land are given as:
  • Cattle: y = 1600 - 11t
  • Sheep: y = 1650 - 30t (where t = 0 refers to 1970).
• In 1970, sheep outnumbered cattle; however, this changed:
  • Set equations equal to find t where sheep numbers drop below cattle.
  • Resulting time approximately t = 2.6, indicating around 1972 or 1973.