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Solving Linear Systems

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Solving Linear Systems

- The solution of a system of equations in two variables is an ordered pair (x,y) that satisfies each equation.
- Consistent – means that there is at least one solution
- Inconsistent - if there are no solutions then the system is inconsistent
- The two lines are parallel

- Dependant – a consistent system with infinitely many solutions
- The two lines are on top of each other

- Independent – a consistent system with exactly ONE solution
- **All this vocabulary only applies to linear systems

Practice and Examples:

- Graph each of the below and classify the system as consistent and independent; consistent and dependant or inconsistent:
- Examples:

Steps:

- 1) Solve for y (put in y=mx+b form) to make graphing easier
- 2) Graph each line by using slope intercept form:
- Find the y-int
- Count off the slope

- 3) Look for places that the graphs intersect and list these as solutions
- 4) Verify solutions algebraically by plugging in each part of the ordered pair solutions

Solving Word Problems

- Word Problems can be solved in the following manner
- Create a linear model for the problem
- Solve each linear equation for the same thing (what we do when we solve for y in the earlier problems) then set these equations equal to each other.
- Examples: pg. 9 29 and 33

Finding Solutions

- Algebraically – solve for one variable, set equal and find solution
- Graphically – graph and look for intersection points
- Simple Examples:
- Solve: a) 2x=8
- b) x-5=2
- c) 2x+5=x-1

Solving Non-Linear Systems

- Just like linear equations are solved by looking for the places that their graphs intersect; non-linear systems are also solved by finding intersection points.
- Examples will be shown later…..

- Solutions or intersection points can be found algebraically by setting each equation equal to each other.
- Example – find the solution(s) for the below set of functions

Practice:

- Find the solution(s) for the following set of functions algebraically:

Solving Non-Linear Systems

- Example 1:
- Use a graph to identify ordered pair solution(s) for the following set of equations

Graph to find the solution for:

- Example 2:

Word Problem Practice

Heather and Amanda each improved their yards

by planting hostas and geraniums. They bought

their supplies from the same store.

Heather spent $187 on 12 hostas and 13 geraniums.

Amanda spent $109 on 4 hostas and 11 geraniums.

What is the cost of one hosta and the cost of one geranium?