Utilizing 6 Methods: Matrices Graphing – Intercepts Graphing – Functions (y = ) Linear...
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Transcript of Utilizing 6 Methods: Matrices Graphing – Intercepts Graphing – Functions (y = ) Linear...
DefinitionA system of 2 linear equations is: 2 linear
equations, both of which contain the same 2 variables.
Example: 5x + 4y = 60 (Equation #1) x + 2y = 18 (Equation #2)
Real-World ExampleMr. K goes to the concession stand at a
baseball game. He buys 5 cheesesteaks and 4 large sodas and pays a total of 60 dollars. Later in the game, Mr. S goes to the concession stand. Mr. S buys 1 cheesesteak and 2 large sodas and pays a total of 18 dollars.
What is the cost of one cheesesteak, and the cost of one large soda?
SolutionTo find a solution to a system of 2 linear
equations means to find the point(s) in common (if possible) between the two equations.
In our example, this means to find the cost of one cheesesteak and the cost of one large soda.
NOTE: Sometimes a solution does not exist, or there are an infinite number of solutions.
Setting Up the SystemTo set-up a system, do the following:1.Identify the variables (unknowns) in the
problem. Example: Let c = Cost of one Cheesesteak
Let s = Cost of one large Soda
2.Write 2 separate equations, using the variables above, to model the situation. Example: Each trip to the concession stand is
an equation:Mr. K’s trip: 5c + 4s = 60Mr S’s trip: c + 2s = 18
Solving the System This tutorial will use 6 different methods to
solve a system of 2 linear equations:1. Matrices2. Graphing – Intercepts3. Graphing – Functions (y =)4. Linear Combination5. Substitution – One Equation into the Other6. Substitution – Equations Equal to Each Other
Solving the SystemMethod 1: MatricesUsing the system of equations:
5c + 4s = 60 c + 2s = 18
Re-write as a matrix equation:
c
s
⎡
⎣⎢⎤
⎦⎥
5
1
⎡
⎣⎢
4
2
⎤
⎦⎥ =
60
18
⎡
⎣⎢
⎤
⎦⎥
Solving the SystemMethod 1: Matrices (Continued)Solve the system using inverse matrices:
Therefore, cost of one cheesesteak is c = $8,and the cost of one large soda is s = $5.
c
s
⎡
⎣⎢⎤
⎦⎥
5
1
⎡
⎣⎢
4
2
⎤
⎦⎥ = 60
18
⎡
⎣⎢
⎤
⎦⎥
c
s
⎡
⎣⎢⎤
⎦⎥
5
1
⎡
⎣⎢
4
2
⎤
⎦⎥= 60
18
⎡
⎣⎢
⎤
⎦⎥
5
1
⎡
⎣⎢
4
2
⎤
⎦⎥
5
1
⎡
⎣⎢
4
2
⎤
⎦⎥
−1 −1
c
s
⎡
⎣⎢⎤
⎦⎥ = 8
5
⎡
⎣⎢⎤
⎦⎥
Solving the SystemMethod 2: Graphing - InterceptsFor each equation:
Find the x-intercept: Point (x,0)Find the y-intercept: Point (0,y)
Let x = the cost of a cheesesteakLet y = the cost of a large sodaThen our system becomes
5x + 4y = 60 x + 2y = 18
Solving the SystemMethod 2: Graphing – Intercepts
5x + 4y = 60 (Equation 1)
X-intercept: 5x + 4(0) = 60 5x = 60 5 5 x = 12 Pt. (12,0)
Y-intercept:5(0) + 4y = 60 4y = 60 4 4y = 15 Pt. (0,15)
x + 2y = 18 (Equation 2)
X-intercept:x + 2(0) = 18x = 18 Pt.
(18,0)
Y-intercept:(0) + 2y = 18
2y = 18 2 2 y = 9 Pt.
(0, 9)
5x + 4y = 60 (Equation 1) x + 2y = 18 (Equation 2)
Solving the SystemMethod 2: Graphing – Intercepts (Cont.)
Step 1: Graph Equation 1 – Using the intercepts
Solving the SystemMethod 2: Graphing – Intercepts (Cont.)
Step 2: Graph Equation 2 – Using the intercepts
Solving the SystemMethod 2: Graphing – Intercepts (Cont.)
Step 2: Graph Equation 2 – Using the intercepts
Solving the SystemMethod 2: Graphing – Intercepts (Cont.)
Step 3: Find the intersection - the solution.
Solving the SystemMethod 2: Graphing – Intercepts (Cont.)
Step 3: Find the intersection - the solution.
Solving the SystemMethod 2: Graphing – Intercepts (Cont.)
The intersection point (8,5) is the solution to the system of equations.
8, the x-coordinate, is the cost of a cheesesteak = $8
5, the y-coordinate, is the cost of a large soda = $5
Solving the SystemMethod 3: Graphing - Functions (y =)Let x = the cost of a cheesesteakLet y = the cost of a large sodaThen our system becomes
5x + 4y = 60 x + 2y = 18
Now, we must solve each equation for “y”
Solving the SystemMethod 3: Graphing - Functions (y =)
5x + 4y = 60 (Equation 1)
-5x -5x 4y = 60 – 5x
4 4
x + 2y = 18 (Equation 2)
-x -x 2y = 18 - x
2 2
y =604
−54x
y =15 −1.25x
y =182
−x2
y =9 −0.5x
Solving the SystemMethod 3: Graphing - Functions (y =)Type each equation into “y =“ in calculator
Set an appropriate “WINDOW”For example: Xmin = -5, Xmax = 20, Xscl = 5
Ymin = -5, Ymax = 20, Yscl = 5, Xres = 1
Press “GRAPH”
Find the intersection point
y1 =15 −1.25xy2 =9 −0.5x
Solving the SystemMethod 3: Graphing - Functions (y =)Using the Window above, graph should look like:
Solving the SystemMethod 3: Graphing - Functions (y =)To find the intersection point:Press “2nd” “TRACE” “5:intersect” “ENTER”Press “ENTER” – 3 more timesSolution is:
Intersection:X = 8 Y = 5The ordered pair (8,5)
8, the x-coordinate, is the cost of a cheesesteak = $8
5, the y-coordinate, is the cost of a large soda = $5
Solving the SystemMethod 4: Linear CombinationFrom the original system of equations:
5c + 4s = 60 (equation 1) c + 2s = 18 (equation 2)
Linear Combination is the process of eliminating one of the variables. This is done by multiplying one (or both) of the equations by a number, so that when adding the equations together, one variable is canceled out.
Example: 2x + (-2x) = 0
Solving the SystemMethod 4: Linear CombinationIn our equations: we can cancel the +4s in equation
1, by multiplying equation 2 by (-2).5c + 4s = 60 5c + 4s = 60
-2 (c + 2s = 18) -2c +-4s = -36 (Add Equations) 3c = 24
(Solve for c) 3c = 24 3 3 c = 8
(Cost of a
cheesesteak)
Now use this to solve for the other variable.
Solving the SystemMethod 4: Linear CombinationYou can substitute c = 8 into either of the original
equations, and solve for the other variable.Equation 1 Equation 2
5c + 4s = 60 c + 2s = 185(8) + 4s = 60 8 + 2s = 1840 + 4s = 60 -8 -8-40 -40 2s = 10
4s = 20 2 2 4 4 s = 5
s = 5So, s, the cost of a soda = $5. And, the solution of: c = 8 and s = 5, satisfies both original equations.
Solving the SystemMethod 5: Substitution – One Equation into the OtherFrom the original system of equations:
5c + 4s = 60 (equation 1) c + 2s = 18 (equation 2)
By solving one equation for one of the variables, and substituting into the other equation, the system becomes one equation with one variable, since there is a one common point (c,s) that satisfies both equations.
So, we need to solve one of our equations for c (or s) and then substitute the result into the other equation.
Solving the SystemMethod 5: Substitution – One Equation into the OtherSolving equation 2 for “c”:
c + 2s = 18 - 2s - 2sc = 18 – 2s (NOTE: This is NOT “16s”)
Now, substitute this new expression for c into equation 1 for c:
5c + 4s = 605(18 – 2s) + 4s = 60
Solving the SystemMethod 5: Substitution – One Equation into the OtherNow, solve this equation for s:
5(18 – 2s) + 4s = 6090 – 10s + 4s = 6090 – 6s = 60
-90 - 90 - 6s = - 30
-6 -6 s = 5
Solving the SystemMethod 5: Substitution – One Equation into the OtherNow, substitute s = 5 into either of the original equations and solve for c.
5c + 4s = 60 (equation 1)5c + 4(5) = 605c + 20 = 60 - 20 - 205c = 405 5
c = 8
So, s, the cost of a soda = $5. And, c, the cost of a cheesesteak = $8.
Solving the SystemMethod 6: Substitution – Equations Equal to Each OtherFrom the original system of equations:
5c + 4s = 60 (equation 1) c + 2s = 18 (equation 2)
By solving each equation for the same variable, and setting the expressions equal to each other, the system becomes one equation with one variable, since there is a one common point (c,s) that satisfies both equations.
So, we need to solve both of our equations for c (or s), and then set them equal to each other.
Solving the SystemMethod 6: Substitution – Equations Equal to Each Other
5c + 4s = 60 (equation 1)
c + 2s = 18 (equation 2)
Solving for c:5c + 4s = 60 - 4s - 4s5c = 60 – 4s5 5 c = 60 - 4s 5 5 OR c = 12 – 0.8s
Solving for c:c + 2s = 18 - 2s - 2sc = 18 – 2s
Solving the SystemMethod 6: Substitution – Equations Equal to Each OtherNow, set the two expression for c equal to each
other and solve for s:12 – 0.8s = 18 – 2s + 0.8s + 0.8s12 = 18 – 1.2s-18 -18-6 = -1.2s-1.2 -1.2 5 = s
Solving the SystemMethod 6: Substitution – Equations Equal to Each OtherNow substitute s = 5 into either of the
expressions found previously for c.c = 18 – 2sc = 18 – 2(5)c = 18 – 10c = 8
So, s, the cost of a soda = $5. And, c, the cost of a cheesesteak = $8.
Verifying the Solution to the SystemThe “Check-Step”All of the methods of solving the system resulted in
the same solution: s = 5, c = 8.To verify this solution satisfies both original
equations, substitute the solution values into the original equations.
Equation 1: Equation 2:5c + 4s = 60 c + 2s = 185(8) + 4(5) = 60 8 + 2(5) = 1840 + 20 = 60 8 + 10 = 18 60 = 60 ✓ 18 = 18 ✓
IMPORTANTIn the example given in this tutorial, one
solution exists. However, not all systems of 2 linear equations have exactly one solution.
It is possible to have either:1. An infinite number of solutions, OR2. No solution
If a system has an infinite number of solutions, this means that the equations are actually the same equation (the same line if graphing), just written in different forms.
If a system has no solutions, this means that the equations have the same slopes, but different y-intercepts (parallel lines if graphing).
IMPORTANTTo determine if one solution exists, write each
equation in Standard Form: Ax + By = C.For a system of 2 linear equations, this would
look like: Ax + By = C (equation 1)
Dx + Ey = F (equation 2)If the system has infinite solutions, then:
If the system has no solution, then:
Otherwise, the system has ONE solution.
A
D=BE
=CF
A
D=BE
≠CF
Practice ProblemsSolve each of the following systems of linear
equations by the method indicated. First, make sure the system does, in fact, have one solution. When finished with each problem, be sure to verify the solution found.
1.) Method: Matrices2x + 3y = 23x – 4y = -14
Practice Problems2.) Method: Graphing - Intercepts
2x + 9y = 362x – y = 16
3.) Method: Graphing – Functions (y=)5x – 6y = 482x + 5y = -3
Practice Problems4.) Method: Linear Combination
4x – 3y = 175x + 4y = 60
5.) Method: Substitution – One Equation into the Other
8x – 9y = 194x + y = -7
Practice Problems6.) Method: Substitution – Equations Equal to
Each Other4x – y = 63x + 2y = 21
7.) Method: You Choose8x – 4y = 234x – 2y = -17
Practice Problems8.) Method: You Choose
-2x + 5y = 9y = 13 - x
9.) Method: You Choose5x + 3y = 12
15x + 9y = 36
Practice Problems10.) Method: You Choose
Suppose that the promotions manager of a minor league baseball team decided to have a giveaway of tote bags and t-shirts to the first 150 fans present. The team owner agrees to a budget of $1350 for the products to be given away. One bag costs $10 and one t-shirt costs $7. How many bags and how many t-shirts should be given away?
Set up the system of 2 linear equations and solve.