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Prerequisite Skills Review

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Title: Solve Systems by Graphing Author: Skip Tyler Last modified by: NBoE Created Date: 2/12/2013 1:29:09 PM Document presentation format: On-screen Show (4:3) – PowerPoint PPT presentation

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Title: Prerequisite Skills Review


1
Prerequisite Skills Review
  • 1.) Simplify 8r (-64r)
  • 2.) Solve 3x 7(x 1) 23
  • 3.) Decide whether the ordered pair (3, -7) is a
    solution of the equation 5x y 8

2
Section 7.1
  • Solving Systems by Graphing

What is a system???? Working with 2 equations
at one time Example 2x 3y 6 X 5y -12
3
What is a system of equations?
  • A system of equations is when you have two or
    more equations using the same variables.
  • The solution to the system is the point that
    satisfies ALL of the equations. This point will
    be an ordered pair.
  • When graphing, you will encounter three
    possibilities.

4
Intersecting Lines
  • The point where the lines intersect is your
    solution.
  • The solution of this graph is (1, 2)

(1,2)
5
Lines intersect one solution
Lines are parallel no solution
Lines coincide infinitely many solutions
6
Parallel Lines
  • These lines never intersect!
  • Since the lines never cross, there is NO
    SOLUTION!
  • Parallel lines have the same slope with different
    y-intercepts.

7
Coinciding Lines
  • These lines are the same!
  • Since the lines are on top of each other, there
    are INFINITELY MANY SOLUTIONS!
  • Coinciding lines have the same slope and
    y-intercepts.

8
What is the solution of the system graphed below?
  1. (2, -2)
  2. (-2, 2)
  3. No solution
  4. Infinitely many solutions

9
Name the Solution
10
Name the Solution
11
Name the Solution
12
How to Use Graphs to Solve Linear Systems
Consider the following system
x y 1 x 2y 5
We must ALWAYS verify that your coordinates
actually satisfy both equations.
To do this, we substitute the coordinate (1 , 2)
into both equations.
x y 1 (1) (2) 1 ?
x 2y 5 (1) 2(2) 1 4 5 ?
Since (1 , 2) makes both equations true, then (1
, 2) is the solution to the system of linear
equations.
13
Solving a system of equations by graphing.
  • Let's summarize! There are 3 steps to solving a
    system using a graph.

Graph using slope and y intercept or x- and
y-intercepts. Be sure to use a ruler and graph
paper!
Step 1 Graph both equations.
This is the solution! LABEL the solution!
Step 2 Do the graphs intersect?
Substitute the x and y values into both equations
to verify the point is a solution to both
equations.
Step 3 Check your solution.
14
1) Find the solution to the following system
  • 2x y 4
  • x - y 2
  • Graph both equations. I will graph using x- and
    y-intercepts (plug in zeros).
  • Graph the ordered pairs.

2x y 4 (0, 4) and (2, 0)
x y 2 (0, -2) and (2, 0)
15
Graph the equations.
  • 2x y 4
  • (0, 4) and (2, 0)
  • x - y 2
  • (0, -2) and (2, 0)
  • Where do the lines intersect?
  • (2, 0)

2x y 4
x y 2
16
Check your answer!
  • To check your answer, plug the point back into
    both equations.
  • 2x y 4
  • 2(2) (0) 4
  • x - y 2
  • (2) (0) 2

Nice joblets try another!
17
2) Find the solution to the following system
  • y 2x 3
  • -2x y 1
  • Graph both equations. Put both equations in
    slope-intercept or standard form. Ill do
    slope-intercept form on this one!
  • y 2x 3
  • y 2x 1
  • Graph using slope and y-intercept

18
Graph the equations.
  • y 2x 3
  • m 2 and b -3
  • y 2x 1
  • m 2 and b 1
  • Where do the lines intersect?
  • No solution!

Notice that the slopes are the same with
different y-intercepts. If you recognize this
early, you dont have to graph them!
19
y 2x 0 y -1x 3
Slope -1/1
Y
Slope 2/1
y-intercept 0
X
Up 2 and right 1
Down 1 and right 1
y-intercept 3
(1,2)
The solution is the point they cross at (1,2)
20
y x - 3 y -3x 1
Slope -3/1
Y
Slope 1/1
y-intercept -3
X
y-intercept 1
The solution is the point they cross at (1,-2)
21
y -2x 4 y 2x 0
Slope 2/1
Y
Slope -2/1
y-intercept 4
X
y-intercept 0
The solution is the point they cross at (1,2)
22
Graphing to Solve a Linear System
Using the slope intercept form of these
equations, we can graph them carefully on graph
paper.
Start at the y - intercept, then use the slope.
Lastly, we need to verify our solution is
correct, by substituting (3 , 1).
23
Practice Solving by Graphing
y x 1 ? (0,1) and (-1,0) y x 3 ?
(0,3) and (3,0) Solution is probably (1,2)
Check it 2 1 1 true 2 1 3
true therefore, (1,2) is the solution
(1,2)
24
Practice Solving by Graphing
Inconsistent no solutions
y -3x 5 ? (0,5) and (3,-4) y -3x 2 ?
(0,-2) and (-2,4) They look parallel No
solution Check it m1 m2 -3 Slopes are
equal therefore its an inconsistent system
25
Practice Solving by Graphing
Consistent infinite sols
3y 2x 6 ? (0,2) and (-3,0) -12y 8x -24
? (0,2) and (-3,0) Looks like a dependant system
Check it divide all terms in the 2nd
equation by -4 and it becomes identical to the
1st equation therefore, consistent, dependant
system
(1,2)
26
Ex Check whether the ordered pairs are solns.
of the system. x-3y -5 -2x3y10
  • (-5,0)
  • -5-3(0) -5
  • -5 -5
  • -2(-5)3(0)10
  • 1010
  • Solution
  • (1,4)
  • 1-3(4) -5
  • 1-12 -5
  • -11 -5
  • doesnt work in the 1st eqn, no need to check
    the 2nd.
  • Not a solution.

27
Ex Solve the system graphically. 2x-2y
-8 2x2y4
(-1,3)
28
Ex Solve the system graphically. 2x4y12 x2y
6
  • 1st eqn
  • x-int (6,0)
  • y-int (0,3)
  • 2ND eqn
  • x-int (6,0)
  • y-int (0,3)
  • What does this mean?
  • the 2 eqns are for the same line!
  • many solutions

29
Ex Solve graphically x-y5 2x-2y9
  • 1st eqn
  • x-int (5,0)
  • y-int (0,-5)
  • 2nd eqn
  • x-int (9/2,0)
  • y-int (0,-9/2)
  • What do you notice about the lines?
  • They are parallel! Go ahead, check the slopes!
  • No solution!

30
What is the solution of this system?
3x y 8 2y 6x -16
  1. (3, 1)
  2. (4, 4)
  3. No solution
  4. Infinitely many solutions

31
You Try It
Graph the system of equations. Determine whether
the system has one solution, no solution, or
infinitely many solutions. If the system has one
solution, determine the solution.
32
Problem 1
The two equations in slope-intercept form are
Plot points for each line.
Draw in the lines.
These two equations represent the same
line. Therefore, this system of equations has
infinitely many solutions .
33
Problem 2
The two equations in slope-intercept form are
Plot points for each line.
Draw in the lines.
This system of equations represents two parallel
lines.
This system of equations has no solution
because these two lines have no points in common.
34
Problem 3
The two equations in slope-intercept form are
Plot points for each line.
Draw in the lines.
This system of equations represents two
intersecting lines.
The solution to this system of equations is a
single point (3,0) .
35
Key Skills
  • Solve a system of two linear equations in two
    variables graphically.

y 2x ? 1
solution (2, 3)
36
Key Skills
  • Solve a system of two linear equations in two
    variables graphically.

y 2x 2
y x 1
solution (1, 0)
37
Key Skills
  • Solve a system of two linear equations in two
    variables graphically.

y 2x 2
y 2x 4
No solution, why?
Because the 2 lines have the same slope.
38
Key Skills
TRY THIS
  • Solve a system of two linear equations in two
    variables graphically.

y 3x 2
solution (-3, -1)
39
Key Skills
TRY THIS
  • Solve a system of two linear equations in two
    variables graphically.

2x 3y -12
4x 4y 4
solution (-1.5, -3)
40
Consider the System
BACK
41
Graph each system to find the solution
(-3, 1)
1.) x y -2 2x 3y -9 2.) x y
4 2x y 5 3.) x y 5 2x 3y
0 4.) y x 2 y -x 4 5.) x -2
y 5
(1, 3)
(????)
(????)
(-2, 5)
42
Check whether the ordered pair is a solution of
the system
1.) 3x 2y 4 (2, -1) -x 3y
-5 2.) 2x y 3 (1, 1) or (0,
3) x 2y -1 3.) x y 3 (-5,
-2) or (4, 1) 3x y 11
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