## 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.

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

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

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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).

- For example:

### 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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## 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!