Algebra 2 Unit 3 Study Guide

Study Notes

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.

Description

This quiz focuses on systems of two linear equations in two variables, including definitions and examples. It tests your understanding of solutions within linear equations, helping reinforce key concepts from Algebra 2. Perfect for reviewing Unit 3 material!