 # Sections 3.1 & 3.2  A collection of equations in the same variables.

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Sections 3.1 & 3.2

 A collection of equations in the same variables.

 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 ordered pair (2, -1) is the solution of the system below.  y = x – 3 y = 5 – 3x

Exploring Graphs of Systems YOU WILL NEED: graph paper or a graphing calculator

System I.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.

I. Y = 2x – 1 Y = -x + 5  Plug in your equations to Y=  Press Graph

 Press 2 nd, CALC  Select 5: INTERSECT

 FIRST CURVE? Press Enter to select the line.  SECOND CURVE? Press Enter to select the 2 nd line  GUESS? Move the cursor close to the point of intersection and press Enter

Intersection Point (2, 3)

 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.

 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)

 If a system does not have a solution, it is called inconsistent. (PARALLEL LINES) (NO SOLUTION)

 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.

 Graph and find the intersection point like Activity 1.  y = 5 – x y = Consistent & Independent

2x + y = 3 3x – 2y = 8 Solve the first equation for y.

 SUBSTITUTE 3 – 2x into the second equation for y. SOLVE

 Substitute 2 for x in either original equation to find y.

 Solution: (2, -1)  Check:

 Involves multiplying and combining the equations in a system in order to eliminate a variable.

 Now plug in y = 1 into either of your two original equations.

 Pg 160-163  Pg 168-170

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