### 3.7_prep

```Visual Algebra for Teachers
Activity Set 3.7
PREP PPTX
Visual Algebra for Teachers
Chapter 3
REAL NUMBERS AND
Visual Algebra for Teachers
Activity Set 3.7
Systems of Equations
PURPOSE
To learn:
 To use the substitution and elimination methods of
solving systems of equations.
 To understand how these solutions can be viewed
graphically
MATERIALS
Graphing calculator with table functions
INTRODUCTION
SYSTEMS OF EQUATIONS
A system of equations is two or more equations of the
same basic type.
A general rule for solving a system of equations is that
you need the same number of equations as “unknowns”
or variables.
That is, you can solve a system of two equations in two
unknowns, a system of three equations in three
unknowns, etc.
There are two main techniques for solving a system of
equations: The Substitution Method and the Elimination
Method.
The Substitution Method
This is a system of two equations in two unknowns
2 m  3n  6
4m  4n  7
1) Solve for one variable in one equation
2 m  3n  6
2 m  6  3n
m 
6  3n
2
S olve for m (this is just one w ay, solving for n also w orks)
The Substitution Method
2 m  3n  6
4m  4n  7
m
6  3n
2
2) Substitute the resulting value into the other equation to
reduce the number of variables in the second equation.
 6  3n
4
 2
4m  4n  7

  4n  7

m
S ubstitute m 
6  3n
2
into second equation
The Substitution Method
Simplify and then solve the new equation for the second
variable (n):
Substitute the value for n
back into either equation
 6  3n 
and solve for m.
4
 4n  7


2


2  6  3n   4 n   7
 19 
2m  3 
6
 2 
12  2 n   7
4 m  3 19   12
2 n  19
n
19
4 m  12  57   45
2
m
45
4
The Substitution Method
n
The final solution to this system is
19
m
2
 45
4
2m

  45 
2

 4 
3n  6
 19 
3
6
 2 
2   45   6 19

4m
 4n
 7
  45 
 19 
4
  4
  7
 4 
 2 
24
 90  114  24
These values create valid equations.
4   45   8 19
   28
 180  152   28
The Elimination Method
Multiply one or more equations by a carefully chosen constant
For example, if we multiply the first equation by -2, we can
“eliminate” m when we add the equations:
1. S ystem
2 . M u ltip ly b y -2
2 m  3n  6
 2  2 m  3n    2  6 
4m  4n  7
4m  4n  7
3. S im plify
 4 m  6 n   12
4m  4n  7
The Elimination Method
4. A dd
5 . S im p lify
4m  6n  12
n
4m  4n  7
19
2
0  2n  19
Just like with Substitution; substitute the value for n back into
either equation and solve for m.
n
19
2
m
 45
4
Question #1
Jack and Jill and Little Boy Blue have competing
and p dollars of profit, the relationship between c and p
can be described using the following equations
For Jack and Jill :
For L ittle B oy B lue :
c  10 p  20
c  5 p  25
Question #1a
Write in your coursepack, Activity Set 3.7 #1a
If Jack and Jill sell 30 cups of lemonade, what is their
profit? (Hint: Use the J & J equation)
If Little Boy Blue sells 30 cups of lemonade, what is his
profit? (Hint: Use the LBB equation)
Question #1b
Write in your coursepack, Activity Set 3.7 #1b
If Jack and Jill make a profit of \$5.00, how many cups of
lemonade did they sell? (Hint: Use the J & J equation)
If Little Boy Blue makes a profit of \$5.00, how many cups
of lemonade did he sell? (Hint: Use the LBB equation)
Question #1cde
Write in your coursepack, Activity Set 3.7 #1cde
c. For the lemonade stands, profit is a function of the cups
of lemonade sold. What type of function is this?
d. Solve each equation for p to determine p as a function
of c. What type of function is this? Is this the same
e. For 0 cups of lemonade, how much profit do Jack and
Jill make? Little Boy Blue? What do these values, the pintercepts, mean in this context?
Question #1fg
Write in your coursepack, Activity Set 3.7 #1fg
f. By looking at the function equations, will the functions
intersect? How can you tell by just looking at the function
equations?
g. Graph the two functions and mark any points of
intersection on the graph. Scale your axes carefully and
label everything clearly. Do negative values for c make
any sense here?
Question #1h
Write in your coursepack, Activity Set 3.7 #1h
Use the Substitution Method to solve the original
system of equations and determine where Jack and Jill
and Little Boy Blue make the same amount of profit.
units (cups, dollars).
Question #1i
Write in your coursepack, Activity Set 3.7 #1i
Use the Elimination Method to solve the original system
of equations and determine where Jack and Jill and Little
Boy Blue make the same amount of profit.