Algebra 2 Unit 3 Study Guide

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Questions and Answers

What is a system of two linear equations in two variables?

Ax + By = C and Dx + Ey = F

A __________ of a system of linear equations in two variables is an ordered pair (x,y) that is a solution of both equations.

solution

Is (1,3) a solution for the system where 5x - 3y = -4 and x + 2y = 7?

Yes

The __________ can be used to approximate the solution of the system.

<p>graph of the system</p> Signup and view all the answers

What is the solution to the linear system -2x + y = 8 and 3x + y = -2?

<p>(-2, 4)</p> Signup and view all the answers

When did the number of grazing sheep fall below the number of grazing cattle according to the model?

<p>Around 1972 or 1973</p> Signup and view all the answers

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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.

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