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

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Solving Linear Systems

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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


Graph to find the solution for

Graph to find the solution for:

  • Example 2:


Solution is 0 2

Solution is (0,-2)


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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