Algebra and Linear Equations Review

Math Studies SL10/30/13

Laws of Exponents

am • an =

am ÷ an =

(am)n =

(ab)n =

(a÷b)n =

a0 =

a-n =

1÷ a-n =

Simplify

(2c3d)4

2 x+2 / 2 x-1

(2a 2 / b 2) 3

(c 3 / d 5) 0

2w4 • 3w

(d 2 • d 7)/ d5

12p 4/(3p 2)

24m2n4/(6m2n)

(k4)5/(k3•k6)

5s5t • 2t3

3xy4 • 2xy -3

a -3 b2 / c -1

(ab) -2

[(b 3) 4 • b 5]

(b 2 • b 6)

2a -1/d 2

(4 + 2x) / x -1

(2x 5+ x2) / x -2

Product of Terms: (a+b)(c+d)

Distribution of terms (FOIL)

Expand (a+b) (c+d)

Expand and Simplify

(x+7) (x+2)

(2x+3) (x – 4)

(3x – 5) (2x+7)

(7x - 3)(4 - 5x)

(x+2) (x2+3x+4)

(4x–1)(x2–x–1)

–x (x+6) (1-x)

(2x+3)(2x-1)(x-2)

(a+b)(c+d+e)

(a+b+c+d)(e+f+g)

Linear Equations

Solve for x: 3(x+2) + 2(x+4) = 19

5(2x+1) - 3(x-1) = -6

5x - 5 = 4x+1

5(2x - 1)+2 = 10x - 3

Linear Equations:Solve using a TI-83

1. Press MATH

2. Go to 0: Solver…

3. Manipulate your equation so that it is equal to 0 You cannot enter equation unless it is set equal to 0

4. Press ENTER

5. Make a guess at the solution and set X = GUESS

6. Adjust bound so that your solution is between {lower bound, upper bound}

7. Press ALPHA + ENTER

8. You will now have X = SOLUTION, if there are multiple solutions repeat the steps and adjust your guess OR change your bounds

Linear Problem Solving

Evan’s father is presently three times as old as Evan. In 11 years’ time his father will be twice as old as him how old is he?

Herman has a collection of 2 cent and 5 cent stamps. He has as many 2 cent stamps as 5 cent stamps and the total value of the stamps is 66 cents. How many 5 cent stamps do he have?

Mad sells lemonade for $1, juice for $1.50, and coffee for $2. On one day the number of coffees she sells is twice the number of lemonades she sells, and 4 more than the number of juices she sells. If she earns a total of $74, how many lemonades did she sell?

Formula Rearrangement

For some situations it may be useful or necessary to rearrange your formula so that it equals a different variable.

For example the equation for Volume = ⅓ π r2

h

Rearrange this equation to find Height as a function of Volume and Radius (we call this making h the “subject” of the equation)

Make r the “subject” of the equation

Systems of Equations

Methods for solving:

Solve

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3x + 4y =1

x − 2y = 7

⎧ ⎨ ⎩

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6x + y =13

2x − 3y =16

⎧ ⎨ ⎩

€

x + 3y =1

-3x + 7y = 21

⎧ ⎨ ⎩

€

x + 4y = −2

-3x + 2y =13

⎧ ⎨ ⎩

Systems of Equations:Solve using a TI-83

You can solve systems of equations of any size using matrices. TI-83s will allow you to skip several steps in the process and give you your solutions if

you do the following:

1. Press SHIFT + x -1 (MATRIX)

2. Press TWICE to get to EDIT

3. Select whichever letter you want and press ENTER

4. Define size of matrix (remember rows by columns)

5. Enter coefficients and solutions in matrix in appropriate locations, after each number press ENTER

6. Press 2nd + MODE (QUIT)

Systems of Equations:Solve using a TI-83

7. Press SHIFT + x -1 (MATRIX)

8. Press ONCE to get to MATH

9. Go down to B: rref( and Press ENTER

10.Press SHIFT + x -1 (MATRIX)

11.Select your matrix and Press ENTER

12.Close parentheses and Press ENTER

13.You should get your solution

Solve using TI-83

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1.4x − 2.3y = −1.3

5.7x − 3.4y =12.6

⎧ ⎨ ⎩

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3.6x − 0.7y = −11.37

4.9x + 2.7y = −1.23

⎧ ⎨ ⎩

€

y = 4.5x − 4.75

x = y +1.3

⎧ ⎨ ⎩

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x + 3y + 4z =19.4

2x −1.5y + 2.1z = 3.75

3.1x + 2y − z =10.71

⎧

⎨ ⎪

⎩ ⎪

Systems of Equations:Using a TI-83 EXAMPLE

System:

Matrix [A]:

rref([A]):

Solution: (3,-2)

€

3x + 4y =1

x − 2y = 7

⎧ ⎨ ⎩

€

3 4 1

1 −2 7

⎡

⎣ ⎢

⎤

⎦ ⎥

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1 0 3

0 1 −2

⎡

⎣ ⎢

⎤

⎦ ⎥

Quadratic Equations

What do we call the graph of a quadratic equation?

What are the key points on the graph? How can you find these points WITHOUT graphing?

Solving Quadratic Equations

How do we solve quadratic equations?

Solve: x2+3x+1=12.44