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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

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Solving linear systems
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
Practice and Examples:

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

  • Examples:


Steps
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
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
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
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
Practice:

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


Solving non linear systems1
Solving Non-Linear Systems

  • Example 1:

    • Use a graph to identify ordered pair solution(s) for the following set of equations


Graph should look like this
Graph should look like this:

Identify Solutions

Check Algebraically




Word problem practice
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?


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