Sections 3.1 & 3.2 A collection of equations in the same variables.
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Transcript of Sections 3.1 & 3.2 A collection of equations in the same variables.
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Chapter 3
Systems of Linear Equations
& Inequalities
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Solving Systems by
Graphing, Substitution, or
EliminationSections 3.1 & 3.2
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A collection of equations in the same variables.
System of Equations
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The solution of a system of 2 linear equations in x and y is any ordered pair, (x, y), that satisfies both equations.
The solution (x, y) is also the point of intersection for the graphs of the lines in the system.
The solution of a system…
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The ordered pair (2, -1) is the solution of the system below.
y = x – 3 y = 5 – 3x
Example, Pg 156
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ACTIVITY 1
Exploring Graphs of
SystemsYOU WILL NEED: graph paper or a graphing calculator
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SystemI. Y = 2x – 1 Y = -x + 5
II. Y = 2x – 1 Y = 2x + 1
III. Y =
Y = x + 2
Graph System I at left. ◦ Are there any points
of intersection?◦ Can you find exactly
one solution to the system? If so, what is it?
Repeat for Systems II and III.
Graphing Systems of Equations
4
38 x
4
3
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I. Y = 2x – 1 Y = -x + 5
Plug in your equations to Y=
Press Graph
Using the Calculator
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Press 2nd, CALC
Select 5: INTERSECT
To find point of Intersection
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FIRST CURVE? Press Enter to select the line.
SECOND CURVE? Press Enter to select the 2nd line
GUESS? Move the cursor close to the point of intersection and press Enter
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Intersection Point(2, 3)
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CLASSIFYING SYSTEMS
OF EQUATIONS
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Graphing a system in 2 variables will tell you whether a solution for the system exists.
3 possibilities for a system of 2 linear equations in 2 variables.
Graphing
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If a system of equations has at least 1 solution, it is called consistent
◦If a system has exactly one solution, it is called independent(INTERSECTING)
◦If a system has infinitely many solutions, it is called dependent (SAME LINE) (COINCIDING)
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If a system does not have a solution, it is called inconsistent. (PARALLEL LINES) (NO SOLUTION)
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Graph and Classify each system. Then find the solution from the graph.
x + y = 5 x – 5y = -7
Begin by solving each equation for y.
Example
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Graph and find the intersection point like Activity 1.
y = 5 – x y =
Consistent & Independent
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USING SUBSTITUTI
ON
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2x + y = 33x – 2y = 8
Solve the first equation for y.
EXAMPLE
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SUBSTITUTE 3 – 2x into the second equation for y. SOLVE
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Substitute 2 for x in either original equation to find y.
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Solution: (2, -1)
Check:
Check your Solution!
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Try This…Check your Answer
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Example
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Solving Systems by ELIMINATION
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Involves multiplying and combining the equations in a system in order to eliminate a variable.
Elimination Method
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Example
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Now plug in y = 1 into either of your two original equations.
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ASSIGNMENTPg 160-163
Pg 168-170