6.1 solving systems by graphing: system of linear equations: two or more linear equations solution...
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6.1 Solving Systems by Graphing:
System of Linear Equations: Two or more linear equations
Solution of a System of Linear Equations:Any ordered pair that makes all the equations in a system true.
Trend line: Line on a scatter plot, drawn near the points, that shows a correlation
Consistent: System of equations that has at least one solution.
1) Could have the same or different slope but they intersect.
2) The point where they meet is a solution
Consistent Independent: System of equations that has EXACTLY one solution.
1) Have different slopes
2) Only intersect once
3) The point of intersection is the solution.
Consistent Dependent: System of equations that has infinitely many solutions.1) Have same slopes
2) Same y-intercepts
3) Each point is a solution.
Inconsistent: System of equations that has no solutions.
1) Have same slopes
2) different y-intercepts
3) No solutions
Remember:
Remember:
GOAL:
SOLVING A SYSTEM BY GRAPHING: To solve a system by graphing we must:
1) Write the equations in slope-intercept form (y=mx+b)
2) Graph the equations
3) Find the point of intersection
4) Check
Ex: What is the solution of the system? Use a graph to check your answer.
2 4
2
x y
y x
SOLUTION:
2 2 y y xx
1) Write the equations in slope-intercept form (y=mx+b)
22 44 yy xx
SOLUTION: 2 y x
2) Graph the equations 2 4 y x
SOLUTION: 2 y x
3) Find the solution 2 4 y x
Looking at the graph, we see that these two equations intersect at the point : (-2, 0)
SOLUTION:
2 y x
4) Check
2 4 y x
We know that (-2,0) is the solution from our graph.
2( ) 40 2
0 4 4 0 0 TRUE
0 2 2 0 0 TRUE
YOU TRY IT: What is the solution of the system? Use a graph to check your answer.
2
2 3
x y
y x
SOLUTION:
32 3 2 yy xx
1) Write the equations in slope-intercept form (y=mx+b)
2 2 y y xx
SOLUTION: 3 2 y x
2) Graph the equations 2 y x
SOLUTION: 3 2 y x
3) Find the solution 2 y x
Looking at the graph, we see that these two equations intersect at the point : (2,4)
SOLUTION:
3 2 y x
4) Check
2 y x
We know that (2,4) is the solution from our graph.
4 22
4 4 TRUE
4 3( 22) 4 6 2
4 4 TRUE
SYSTEM WITH INFINITELY MANY SOLUTIONS: Using the same procedure we can see that sometimes the system will give us infinitely many solutions (any point will make the equations true).Ex: What is the solution to the system? Use a graph. 2 2
11
2
y x
y x
SOLUTION:
112
23
y xy x
1) Write the equations in slope-intercept form (y=mx+b)
12 2 12 2
2 y x y xy x
SOLUTION:
11
2
y x
2) Graph the equations
11
2
y x
Notice: Every point of one line is on the other.
SYSTEM WITH NO SOLUTIONS: Using the same procedure we can
see that sometimes the system will give us infinitely many solutions (any point will make the equations true).
Ex: What is the solution to the system? Use a graph. โ๐ ๐+๐=๐
๐ ๐โ๐=๐
SOLUTION:1) Write the equations in slope-intercept form (y=mx+b)
โ๐ ๐+๐=๐โ ๐ฒ=๐๐ฑ+๐
๐ ๐โ๐=๐โโ ๐=โ๐ ๐+๐
SOLUTION: 2) Graph the equations
Notice: These lines will never intersect. NO SOLUTIONS.
๐=๐ ๐+๐
๐=๐ ๐โ๐
WRITING A SYSTEM OF EQUATIONS: Putting ourselves in the real world,
we must be able to solve problems using systems of equations.
Ex: One satellite radio service charges
$10.00 per month plus an activation fee of $20.00. A second service charges $11 per month plus an activation fee of $15. For what number of months is the cost
of either service the same?
SOLUTION: Looking at the data we must be able to do 5 things:1) Relate- Put the problem in simple terms.
Cost = service charge + monthly dues 2) Define- Use variables to represent change:Let C = total Cost Let x = time in months
SOLUTION: (continue)
3) Write- Create two equations to represent the events.
Satellite 1: C = $10 x + $20
4) Graph the equations: Remember to put them in slope/intercept form (y = mx + b)
The two equations are already in y=mx+b form.
Satellite 2: C = $11 x + $15
SOLUTION: 4) Continue๐=๐๐๐+๐๐ ๐=๐๐๐+๐๐
Cost
1 2 3 4
102030
40
5060708090
100
5 6
Months
SOLUTION: 5) Interpret the solution.
Notice: These lines intersect at at (5, 70). Co
st
1 2 3 4
102030
40
5060708090
100
5 6
This means that the two satellite services will cost the same in 5 months and $70.
Months
๐=๐๐๐+๐๐๐=๐๐๐+๐๐
YOU TRY IT: Scientists studied the weights of
two alligators over a period of 12 months. The initial weight and growth rate of each alligator are shown below. After how many months did the two alligators weight the same?
SOLUTION: Looking at the data, Here are the 5 things we must do:1) Relate- Put the problem in simple terms.
Total Weight = initial weight + growth per month.
2) Define- Use variables to represent change:Let W = Total weight Let x = time in months
SOLUTION: (continue)
3) Write- Create two equations to represent the events.
Alligator 1: W = 1.5x + 4
4) Graph the equations: Remember to put them in slope/intercept form (y = mx + b)
The two equations are already in y=mx+b form.
Alligator 2: W= 1.0 x + 6
SOLUTION: 4) ContinueW ๐=๐ ๐+๐
Wei
ght
1 2 3 4
123
4
56789
10
5 6
Months
SOLUTION: 5) Interpret the solution.
Notice: These lines intersect at at (4, 10) This means that the two Alligators will Weight 10 lbs after 4 months.
Months
Wei
ght
1 2 3 4
123
4
56789
10
5 6
VIDEOS: Solve by Graphing
https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/fast-systems-of-equations/v/solving-linear-systems-by-graphing
CLASSWORK:
Page 363-365
Problems: As many as needed to master the
concept.