Number operation review 9 docx

Notice that each successive term is found by multiplying the prior term by 2.. Since each term is multiplied by a constant number 2, there is a constant ratio between the terms.. Sequenc

Practice Question

What is the equation represented in the graph above?

a y  x2 3

b y  x2 3

c y  (x  3)2

d y  (x  3)2

e y  (x  1)3

Answer

b This graph is identical to a graph of y  x2except it is moved down 3 so that the parabola intersects the

y-axis at 3 instead of 0 Each y value is 3 less than the corresponding y value in y  x2, so its equation

is therefore y  x2 3

x

y

1 2 3 4 5 6 1

2 3 4 5

–1 –2

–6 –5 –4 –3

–1 –2 –3

–4

–5

–6

x

y

1 2 3 4 5 6 7 1

2 3 4 5

–1 –2 –3

–1 –2 –3

–4

–5

–6

–7

 R a t i o n a l E q u a t i o n s a n d I n e q u a l i t i e s

Rational numbers are numbers that can be written as fractions (and decimals and repeating decimals) Similarly,

rational equations are equations in fraction form Rational inequalities are also in fraction form and use the

sym-bols <, >,≤, and ≥ instead of 

Example

Factor the top and bottom:

 30

You can cancel out the (x  5) and the (x  2) terms from the top and bottom to yield:

x 7  30

Now solve for x:

x 7  30

x 7  7  30  7

x 23

Practice Question

a. 16

b.13

c. 8

d 2

e 4

Answer

e. To solve for x, first factor the top and bottom of the fractions:

 17

 17

You can cancel out the (x  8) and the (x  2) terms from the top and bottom:

x 13  17

Solve for x:

x 13  13  17  13

x 4

(x + 8)(x + 13)(x 2)

(x + 8)(x 2)

(x + 8)(x2+ 11x 26)

(x2+ 6x 16)

(x + 8)(x2+ 11x 26)

(x2+ 6x 16)

(x + 5)(x + 7)(x 2)

(x + 5)(x 2)

(x + 5)(x2+ 5x 14)

(x2+ 3x 10)

 R a d i c a l E q u a t i o n s

Some algebraic equations on the SAT include the square root of the unknown To solve these equations, first iso-late the radical Then square both sides of the equation to remove the radical sign

Example

5c  15  35

To isolate the variable, subtract 15 from both sides:

5c  15  15  35  15

5c  20

Next, divide both sides by 5:

55c 250

c  4

Last, square both sides:

(c)2 42

c 16

Practice Question

If 6d  10  32, what is the value of d?

a 7

b 14

c 36

d 49

e 64

Answer

d To solve for d, isolate the variable:

6d  10  32

6d  10  10  32  10

6d  42



d  7

(d)2 72

d 49

42

6 6d

6

 S e q u e n c e s I n v o l v i n g E x p o n e n t i a l G r o w t h

When analyzing a sequence, try to find the mathematical operation that you can perform to get the next number

in the sequence Let’s try an example Look carefully at the following sequence:

2, 4, 8, 16, 32,

Notice that each successive term is found by multiplying the prior term by 2 (2  2  4, 4  2  8, and so on.) Since each term is multiplied by a constant number (2), there is a constant ratio between the terms Sequences

that have a constant ratio between terms are called geometric sequences.

On the SAT, you may be asked to determine a specific term in a sequence For example, you may be asked

to find the thirtieth term of a geometric sequence like the previous one You could answer such a question by writ-ing out 30 terms of a sequence, but this is an inefficient method It takes too much time Instead, there is a for-mula to use Let’s determine the forfor-mula:

First, let’s evaluate the terms

2, 4, 8, 16, 32,

Term 1  2

Term 2  4, which is 2  2

Term 3  8, which is 2  2  2

Term 4  16, which is 2  2  2  2

You can also write out each term using exponents:

Term 1  2

Term 2  2  21

Term 3  2  22

Term 4  2  23

We can now write a formula:

Term n 2  2n 1

So, if the SAT asks you for the thirtieth term, you know that:

Term 30  2  230  1 2  229

The generic formula for a geometric sequence is Term n  a1 r n 1, where n is the term you are looking for, a1is the first term in the series, and r is the ratio that the sequence increases by In the above example, n 30

(the thirtieth term), a1 2 (because 2 is the first term in the sequence), and r  2 (because the sequence increases

by a ratio of 2; each term is two times the previous term)

You can use the formula Term n  a  r n 1when determining a term in any geometric sequence

Practice Question

1, 3, 9, 27, 81,

What is the thirty-eighth term of the sequence above?

a 338

b 3  137

c 3  138

d 1  337

e 1  338

Answer

d 1, 3, 9, 27, 81, is a geometric sequence There is a constant ratio between terms Each term is three

times the previous term You can use the formula Term n  a1 r n 1to determine the nth term of

this geometric sequence

First determine the values of n, a1, and r:

n 38 (because you are looking for the thirty-eighth term)

a1 1 (because the first number in the sequence is 1)

r 3 (because the sequence increases by a ratio of 3; each term is three times the previous term.) Now solve:

Term n  a1 r n 1

Term 38  1  338  1

Term 38  1  337

 S y s t e m s o f E q u a t i o n s

A system of equations is a set of two or more equations with the same solution If 2c  d  11 and c  2d  13 are presented as a system of equations, we know that we are looking for values of c and d, which will be the same

in both equations and will make both equations true

Two methods for solving a system of equations are substitution and linear combination.

Substitution

Substitution involves solving for one variable in terms of another and then substituting that expression into the second equation

Example

Here are the two equations with the same solution mentioned above:

2c  d  11 and c  2d  13

To solve, first choose one of the equations and rewrite it, isolating one variable in terms of the other It does not matter which variable you choose

2c  d  11 becomes d  11  2c

Next substitute 11  2c for d in the other equation and solve:

c  2d  13

c  2(11  2c)  13

c  22  4c  13

22  3c  13

22  13  3c

9  3c

c 3

Now substitute this answer into either original equation for c to find d.

2c  d  11

2(3)  d  11

6  d  11

d 5

Thus, c  3 and d  5.

Linear Combination

Linear combination involves writing one equation over another and then adding or subtracting the like terms so that one letter is eliminated

Example

x  7  3y and x  5  6y

First rewrite each equation in the same form

x  7  3y becomes x  3y  7

x  5  6y becomes x  6y  5.

Now subtract the two equations so that the x terms are eliminated, leaving only one variable:

x  3y  7

 (x  6y  5) (x  x)  ( 3y  6y)  7  (5)

3y 12

y 4 is the answer

Now substitute 4 for y in one of the original equations and solve for x.

x  7  3y

x 7  3(4)

x 7  12

x 7  7  12  7

x 19

Therefore, the solution to the system of equations is y  4 and x  19.