Download Read : Systems of Equaions

Directions
• Read the following Power Point. Make sure
you understand what a system of equations
is and how to find the solution(s) from the
graph. Make sure you also know how to
determine if an ordered pair is a solution to
a system of equation.
• QUIZ next class!!
Copyright © 2007 Pearson Education, Inc.
Slide 7-1
Systems of Equations
• A set of equations is called a system of equations.
• The solutions must satisfy each equation in the
system.
• If all equations in a system are linear, the system is a
system of linear equations, or a linear system.
Copyright © 2007 Pearson Education, Inc.
Slide 7-2
Graphing Systems of
Equations
• A system is a group of two or
more equations.
• The SOLUTION to the
system is the point that
satisfies ALL the equations.
Copyright © 2007 Pearson Education, Inc.
Slide 7-3
Systems of Equations
• What do solutions to linear
equations look like?
• The solution to the system
will be an ordered pair.
Copyright © 2007 Pearson Education, Inc.
Slide 7-4
Systems of Equations
• Every line has an infinite number of points on it.
• Every point on a line satisfies the equation for that line.
• If two lines intersect they have one point in common.
That point is the solution to the system because it
satisfies both equations.
• The point that is the solution will be on both lines.
• If two lines do not intersect they have no points in
common therefore the system has no solution. (NS)
• If two lines lie on top of each other (coincide) then they
have every point in common so the system has an
infinite number of solutions. (IS)
Copyright © 2007 Pearson Education, Inc.
Slide 7-5
Systems of Equations
• There are several ways to solve systems:
algebraically (substitution and elimination)
graphically
calculator
• We will focus on solving by
graphing.
Copyright © 2007 Pearson Education, Inc.
Slide 7-6
Solving Systems of Equations
• If the lines cross once, there
will be one solution.
• If the lines are parallel, there
will be no solutions.
• If the lines are the same,
there will be an infinite
number of solutions.
Copyright © 2007 Pearson Education, Inc.
Slide 7-7
Linear System in Two Variables
•
Three possible solutions to a linear system in two
variables:
1. One solution: coordinates of a point,
2. No solutions: inconsistent case,
3. Infinitely many solutions: dependent case.
Copyright © 2007 Pearson Education, Inc.
Slide 7-8
Systems of Linear Equations in Two Variables
Solving Linear Systems by Graphing.
There are three possible solutions to a system of linear equations in two
variables that have been graphed:
• 1) The two graphs intersect at a single point. The coordinates give the
solution of the system. In this case, the solution is “consistent” and the
equations are “independent”.
• 2) The graphs are parallel lines. (Slopes are equal) In this case the system
is “inconsistent” and the solution set is empty or null.
• 3) The graphs are the same line. (Slopes and y-intercepts are the same) In
this case, the equations are “dependent” and the solution set is an infinite
set of ordered pairs.
9
Solving systems of
Equations
• Find the solution to the
following system:
2x + y = 4
x-y=2
10
Solving Systems of
Equations
1) Solve each equation for y and graph them. (If you
prefer to graph from standard form you can.)
2x + y = 4  y = -2x + 4
x-y=2y=x-2
2) Find the point where the graphs intersect. You can
graph this in the calculator. After you graph it push
2nd, calc., 5(intersection) Put the cursor on the first
line and push enter. Put the cursor on the second
line and push enter. Push enter once again and the
calculator will give you the point of intersection.
11
y = -2x + 4
y=x-2
x
y
12
Solving Systems of
Equations
• The two lines cross at the
point (2, 0). So the solution
to the system is (2, 0).
• To check your solution,
plug it into each equation
and see if you get a true
statement.
13
Checking the solution to a
system of equations.
2x + y = 4  2(2) + 0 = 4  4 = 4 yes
x - y = 2 2 – 0 = 2  2 = 2 yes
Since the ordered pair works in both equations our solution
is correct.
14
Deciding whether an ordered pair is a
solution of a linear system.
The solution set of a linear system of equations contains all
ordered pairs that satisfy all the equations at the same time.
• Is the ordered pair a solution of the given system?
2x + y = -6
.
x + 3y = 2
Substitute the ordered pair into BOTH equations! It must satisfy both
equations to be a solution to the system.
A) (-4, 2)
2(-4) + 2 = -6
(-4) + 3(2) = 2
-6 = -6
2=2
 Yes
B) (3, -12)
2(3) + (-12) = -6
(3) + 3(-12) = 2
-6 = -6
-33  -6
 No
15