Reminder: Word Problem Stratigies (mixtures/investments)

Ok, I can see loads of people are having trouble with some of the word problems. I was having problems with the fertilizer questions until I went back in my notes and realized the antifreeze example/question we went through in class is exactly the same. Then I experimented with the other questions in the same fashion and It wasn't too hard. I'll go through the antifreeze question (that we all should have in your notes).

Ex) Antifreeze A is 18% alcohol. Antifreeze B is 10% alcohol. How many liters of each should be mixed to get 20L of a mixture of 15% alcohol?

Step 1: Create Let Statements (be specific!)
Let a = the number of liters required of the 18% solution
Let b = the number of liters required of the 10% solution

Ask yourself... "what am I solving for?"

Step 2: Create two equations with two variables in each
L1: a + b = 20
L2: 0.18a + 0.10b = 0.15(20)

The second statement is basically (based on the let statements):
type of the first solution times liters of that solution + type of the second solution times liters of that solution = type of mixture you want times liters of the solution you want.

Maybe thinking of it like that will help y'all (I'm sure that's how Mr. Max has been saying it), instead of... point eighteen times 'a' plus point ten times 'b' equals point fifteen times twenty.

Step 3: Solve
The easiest way would be to throw it in graphmatica, but I didn't have graphmatica handy so the next easiest way was within my calculator (let's just go down the line from easiest to hardest).

I'm going to isolate b (the number of liters required of the 10% solution).
L1: b = 20 - a
L2: b = (3 - 0.18a)/0.10

I got L2 this way:
0.18a + 0.10b = 0.15(20)
0.18a + 0.10b = 3
0.18a + 0.10b - 0.18a = 3 - 0.18a
0.10b/0.10 = (3 - 0.18a)/0.10
b = (3 - 0.18a)/0.10

Remember ;)
2nd, CALC, 5: (intercept), ENTER * 3

Step 4: Create a statment to summarize your answer.
In order to create 20L of 15% alcohol antifreeze, mix 12.5L of 18% with 7.5L of 10% antifreeze.

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All raw information came from our lesson in class and out of my keyboard


:)

Systems Of Linear Equations

In this unit it was explained that 2 or more equations that are on the same plane are PARALLEL, INTERSECTING, OR SUPERIMPOSED.
Also the use of 3 solving methods was explained. The 3 being GRAPHING, SUBSTITUTION, AND ELIMINATION.
The graphing method uses the "y=mx+b", "Ax+By+C=0", And the "y-y1=m(x-x1) formulas, and even the use of the graphing calculator.
The substitution method has 3 steps that help you answer the equations.
Step 1 tells you to put line 1and line 2 into equations.
Step 2 tells you to solve.
Step 3 tells you to put you solution into sentance form.

ex: L1: X+Y=-1
L2:X-2Y=8

Step 1: L1: X=(-1+Y)
L2: (-1+Y)+2Y=8
Step 2: (-1+Y)+2Y=8
-1+3Y=8
3Y=9
Y=3
Step 3: Since Y=3, and since X+2Y=8
Than X+2(3)=8
X+6(-6)=8(-6)
X=2
The solution is therefore (2,3).

The 3rd method, elimination, has 6 to follow.
Step 1 tells you to line up the variables, and the coefficients.
Step 2 tells you to choose a variable and solve for it.
Step 3 says to use algebra to multiply or divide the equation by a value to create Sames or Opposites.
Step 4 explains how to either add or subtract the lines.(Sames are subtracted. Opposites are added.)
Step 5 tells you to solve for the remaining variable.
Step 6 tells you to put you solution into sentance form.

ex: L1:5x+y=16(x3)
15x+3y=48
L2: 2x-3y=3
(add)
(17x)/17=51/17
x=3

And since 5x+y=16
5(3)+y=16
15x(-15)+y=16(-15)
y=1
The solution is (3,1).

This unit also explained how to solve word problems using 4 steps, and any method you choose.
Step 1 tells you to make 2 "Let" statements.
Step 2 tells you to make 2 equations using the 2 variables you created in the first step.
Step 3 tells you to do the math by how ever you find easy.
Step 4 tells you to make a statement to summarize your answer.

Types of Linear Systems

~ Consistent System ~ ordered pair solution
lines intersect; meet at one point
slope of lines are different
intercepts are different unless intersection is on the
intercept



~ Inconsistent System ~ no solution
lines are parallel
slopes are the same
intercepts are all different



~ Dependent System ~ many solutions (infinitely)
lines are super imposed
slopes are the same
intercepts are the same




~ Slope Form ~
y = m(x) + b

~ Standard Form ~
A(x) + B(y) + C = 0

~ Point-Slope Form ~
y - y1 = m(x - x1)