1. Exam 1: Tuesday Sept 14
Exam will cover three weeks of class including today.
Covers written homework, online homework, and labs.
Practice exams are on the website.
Thursday-Friday: Recitation in your lab rooms
Covers s2.4&s2.4 homework.
Bring a review question on a notecard
Monday: Homework due.
Written homework due at 6pm.
Online homework due at 11:59pm.
Monday and Tuesday: Exam review in your recitation rooms.
Notecard questions will be addressed here.
Tuesday: Exam: 7:15pm-8:15pm.
Room assignments on the course website.

2. Solving a system of equations
Finding all values of all variables that make all the equations true.
Ex:
6a+3b=12
4a-15=12b+2
I want to find all pairs (a,b) that make both equations true at the same time.

3. Solving a system of equations
This is exactly the same as solving one equation! You can use the same tools.
Your tools are:
The = sign
Doing the same thing to both sides. (Usually adding).

4. Substitution Method Fancy way of saying: using the meaning of =. Ex:
5x-3y=12
y=4x+1 This means that anywhere I see a ‘y’ I can write “(4x+1)” The parentheses are important!

5. Solving with substitution
5x-3y=12
y=4x+1
5x-3(4x+1)=12
5x-12x-3=12
-7x-3=12
+3=+3
-7x=15
x=-15/7 anywhere I see an ‘x’ I can write “(-15/7)”
y=4(-15/7)+1=(-60/7)+(7/7)=-53/7
(x,y)=(-15/7,-53/7)

6. Elimination Method
Fancy way of saying “adding the same thing to both sides.”
Ex: Ex:
-7x-3= x-3y=12
+3= x+3y=3
-7x = x =15

7. Solving with elimination
-7x-3y=12
2x+3y=3
-5x =15
x=-3  anywhere I see an x, I can write “(-3)”
2(-3)+3y=3
-6+3y=3
=+6
3y=9
y=3
(x,y)=(-3,3)

8. What if I can’t use either method?
Ex:
6a+3b=12
4a-15=12b+2
Solution: Use your equation solving tools (doing the same thing to both sides) to put the equations in a shape where one method will work.
Use whichever method you like best. You only ever need one.

9. Setting up for substitution
6a+3b=12
4a-15=12b+2
6a+3b=12  pick one equation to reshape
-3 = -3b
6a =12-3b
* (1/6) = (1/6)
a=(1/6)(12-3b)
a=2-b/2  now it’s in a shape where I can use substitution.
4a-15=12b+2  finish solving using substitution from here.
a=2-b/2  anywhere I see an “a” I can write “(2-b/2)”
4(2-b/2)-15=12b+2
Etc…
(a,b)=(65/28, -9/14)  final answer

10. Setting up for elimination
6a+3b=12
4a-15=12b+2
6a+3b=12  pick an equation to reshape.
* (2/3) = -(2/3)
-4a-2b=-8
4a-15 =12b+2  now I can add both equations together.
-15-2b=12b-6
Etc….
(a,b)=(65/28, -9/14)  final answer

11. You can use any method you want
If you can get your equations in the right shape, substitution will always work.
If you can get your equations in the right shape, elimination will always work.
Pick the one you are most confident with and stick with it.

12. A jewelry maker has total revenue for necklaces given by R(x)=90
A jewelry maker has total revenue for necklaces given by R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the number of necklaces a and sold. How many necklaces must be produced and sold in order to break even?
70 necklaces
71 necklaces
72 necklaces
73 necklaces
Don’t know

13. solution R(x)=90.75x C(x)=24.50x+4770 Break even when R(x)=C(x)
90.75x=24.50x+4770 hidden substitution
-24.50x=-24.50x
66.25x=4770
x=72. (C:72).

14. Cannot be determined from the information given
Suppose demand is given by p+2q=200, and supply is given by p-5q=60. Find the equilibrium point (in other words, find the solution to the system).
p=$60, q=20 units
p=$120, q=60 units
p=$160, q=20 units
p=$20, q=160 units
Cannot be determined from the information given

15. solution
p+2q=200
-p+5q=-60
 I used elimination because it’s faster here.
7q=140
q=20  use a substitution.
p+2(20)=200
p+40=200
p=160
(p,q)=(160,20) (C)

16. Inequalities
A statement that some number is larger than some other number.
Ex:
x+3>12
Means “the number x+3 is bigger than the number 12”
That leaves open a lot of possibilities for the value of “x+3”!

17. x+3>12
X+3 could be: 13 14
2000
12.5
Etc.

18. 5.25x >1150 5.25x+1150>0 5.25x+1150=0 Both A & B
A plumbing store’s monthly profit from the sale of PVC pipe can be described by P(x)=5.25x-1150 dollars, where x is the number of feet of PVC pipe sold. Set up an inequality that would help determine what level of monthly sales is necessary to incur positive profit.
5.25x >1150
5.25x+1150>0
5.25x+1150=0
Both A & B
All of the above

19. Solution
P(x)=5.25x-1150
The profit is represented by P(x). So a positive profit is when P(x)>0
5.25x-1150>0
If I add the same thing to both sides, I get
5.25x>1150 (A)
Or think of it this way. 5.25x is how much I earn when I sell pipe is how much it costs me to make pipe. I make a profit when what I earn is more than my cost.

20. Inequalities
How does doing the same thing to both sides work for an inequality?
Let’s use our example of x+3>12

21. In order for x+3>12 to be true what has to be true about x?

22. Not always safe Beware multiplying or dividing by a negative!
Ex: -4x+3>1

23. -4x+3>1

24. -4x>-2

25. -x>-2/4

26. X<1/2
NOT x>1/2 !

27. How to deal with negative signs in inequalities
Option 1: Avoid them.
-4x+3>1
+4x=+4x
3>1+4x
Option 2: Flip the inequality
-x>-1/2
x<1/2  drawing a diagram helps!

28. Last thought Option 1 and Option 2 are the same thing: -x>-1/2
+x = +x
0 >x-1/2
+1/2=+1/2
1/2>x
x<1/2