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equations Flashcards

1
Q

If x³ +b³ = (x+b) (x²-xb + 9) and b>0 , what is the value of b

A

3

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2
Q

x³ (x² - 10) = -9x

if x>0, what is the solutions

A
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3
Q

c(ax−b)−d(b−ax)
Which of the following is equivalent to the expression above?

(ax-b)(c+d)

(ax-b)² (c+d)

(b-ax)²(c-d)
(ax-b)(b-ax)(c-d)

A
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4
Q

numberx^4 - number

This means, when an equation does’nt have a number with x , the factored form is always:

256a^4 - 1

A

(ax +b) (ax-b) (cx +b) ²

4x + 1) (4x - 1) (16x² +1

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5
Q

(y² - 32y +256 - k) (y² -32y+256 + k) = (y-16)^4 - 100

what is the value of k

A

(y² - 32y +256 ) = (y-16)²

Therefore, - k + k = 100

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6
Q

s³t³ + 1

a) (st-1) ( s²t²+st + 1)
b) (st- 1 ) (s²t² - st - 1)
c) (st-1) ( (st)²-st + 1)
c) (st+1) ( (st)²-st + 1)

A

1) Prove one equation and analyze how to get the correct answer

a) (st-1) ( s²t²+st + 1) : s³t³ +s²t² +st - s²t² - st - 1 -> s³t³ -1
“1” need to be +1 -> So, the last 1 should be both positives

c) (st+1) ( (st)²-st + 1)

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7
Q

w= 6+ 3xy/75

The equation above gives the quantity w in terms of the quantities x and y Which of the following equations correctly expresses yin terms of w and x?

a) y= 25w + 150 / x
b) y= 25w - 150 / x
c) ) y= 25w -2 / x
d) a) 25w + 2 / x

A

w= 6+ 3xy/75
(w - 6 ) 75 = 3xy
(75w - 75.6)/3 = xy
25w - 150) /x = y

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8
Q 
y-b / x-a = m 
a=
a) -mx + y - b /m
b) mx + y - b /m
c) y - b /m
d) -y +b /m
A
  1. Eliminate the denominator first
    y-b = m (x-a) -> y-b = mx - ma
  2. Isolate a
    y-b-mx/ m = -a
  3. Multiplies -1 , so a be positive
    (y-b+mx/ m) . -1 = a => - y-b+mx/ m
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9
Q

(1+1 /2) . -1 : the -1 multiplies only the fraction as a whole or also affects the sum inside “1+1”

a(1-x) + 5y = 18

A

MULTIPLIES THE FRACTION AS A WHOLE
18 - 5y = a(1-x)
18-5y/a - 1 = -x

(18-5y/a - 1). -1 = x => - 18-5y/a + 1

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10
Q

T= √(uqx)² + (To)²
Which of the following equations correctly gives x in terms of T, To, u?

a) x = T-To/uq
b) x = T-(To)²/uq
c) x= √T²-(To)²/uq

A

c) x= √T²-(To)²/uq

T= √(uqx)² + (To)² => T= (uqx)² + (To)²
√T - To² = √(uqx)²

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11
Q

v = 331.3 √1 + T/ 273.15

√ extends to the T fraction

A

(v / 331.3 - √1 )² = + (√T/ 273.15 )²
(v²/ 331.3² - 1 ) = + T/ 273.15
(v²/ 331.3² - 1 ) 273.15 = + T

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12
Q

(x-h)² + ( y - k)² = r²

a) y = r-x + h + k
b) y = r - (x-h)² + k
c) y = +- √r² - (x-h)² + k

A

r² - (x-h)² = (y-k)²
+ - √r² - (x-h)² = √(y-k)²
+ - √r² - (x-h)² + k = y

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13
Q

k² = m² + n²

a) m = k - n
b) m = √k-n
b) m = √k²-n²

A

b) m = √k-n

the square extends to k-n

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14
Q

The area, A, of the unshaded circular ring shown at left is given by the equation A= π(R²-r^²)
where R is the radius of the larger circle, and r is the radius of the smaller circle. Which of the following equations correctly gives R in terms of A and r?

a) R= √A / pi + r²
b) R= √A / pi + r² ( R= √extends to r²
c) R= √A / pi - r² ( R= √extends to r²

A

A/pi + r² = R²

√A/pi + r² =R B

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15
Q

n = (1.96o / E) ²

a) o = √En/1.96
B) o = E. √n/1.96
C) o = 1.96n /E

A

nE² /1.96²= o²
Simplify

√nE²/√1.96² = √o² -> √nE/1.96

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16
Q

√10X - 2xy = 4x

√ extends to -2xy

a) y = - 18x
b) y = -8x + 5
c) y = -2x + 5
d) -16x² - 5x

A

1) Eliminate the ratio
(√10X - 2xy)² = (4x )² -> 10x - 2xy = 16x²

2) Simplify by dividing 2
5x - xy = 8x²

3) Isolate the result
(8x² - 5x ) / - 1 x = -8x² + 5

17
Q

let’s say we have A and B in the equation. But, there are two of B in the fraction position

R = C-P/ P

A

1) Eliminate the denominator by multiplying it both sides
RP = C- P

2) Again, eliminate the B ( who has a twin in the equation ) by adding B both sides
RP + P = C

3) Isolate the common : P ( 1+ R) = C
4) Isolate the desired one :P = C / (1+R)

18
Q

let’s say we have A and B in the equation. But, there are two of B in the fraction position

C= ab / a + b
a =

A

1) Eliminate the denominator by multiplying both sides
C ( a+b) = ab -> Ca + Cb = ab

2) Eliminate the one that has twin and it’s going to be the result (f(x)
Cb = ab - Ca

3) Isolate the common
Cb = a (b-C)
4) Isolate the result
Cb/ b - C = a

19
Q

1/f = 1/o + 1/i

o =

A

1/f - 1/i = 1/o ->
2) MMC
i - f /fi = 1/o

3) Eliminate the numerator 1 by inverting the ratio : potent ^-1

(i - f /fi)^-1= (1/o)^-1 => fi/ i - f = o

20
Q

V = C ( 1-r)^t

r =

A

V/ C - 1 ^t = -r^t
(V/ C - 1 ). -1^t = r^t

RATIONALIZE THE ^t
^t√- V/ C + 1 = ^t√ r^t => (- V/ C)^1/t + 1 = r

21
Q

L = Lo/x
x = 1/ √ 1 - v²/c²

Extends up to -v²/c²

v=

A

L = Lo/ 1/ √ 1 - v²/c²

INVERT 1/ √ 1 - v²/c²
L = Lo. √ 1 - v²/c²

L/Lo = √ 1 - v²/c²
ELIMINATE THE RATIO , by ²

22
Q

T = √(ugx)²+ (To)²
√ extends to To)²

x=

A

√T² - To²/ug = x

√extends to To²

23
Q 
If f(x)=x²-6x and g(f(x))=-2x²+12x+5
 what is the value of g(-9)?
A

1) f(x) = - 9 : x² - 6x = -9
x= 3

2) -2.3²+ 12.3 + 5 = -5 + 41 = 23

24
Q

If f(x)= 7x -1 and f(g(x)) = 14x + 6, what equartion is equal g(x)

A

7 ( g(x) ) - 1 = f(g(x)) = 14x + 6
7. g(x) - 1 = 14x + 6
7. g(x) = 14x + 7
g(x) = 14x + 7 : 7 = 2x + 1

25
Q

If f(x) = 5x +2 and f(g(x) ) = 5x² + 5x - 3 , what equation is equals to g(x)

A 
f(g(x)) = 5.g(x) + 2 = 5x² + 5x - 3 
5 g(x) = 5x² + 5x - 5 : 5
g(x) = x² + x - 1
26
Q

If f(g(x) ) = 9x² and g(x)= 3x² - 2 . What equation is equals to f(x)

√3x+ √2

√3x + 2

3x+2

3x+6

A

SINCE ALL THE ANSWERS ARE LINEAR, DO IT LINEAR: f(x) = ax+ b
a ( 3x² -2 ) + b = 3x²a - 2a + b = 9x²

2) COMPARE
3x²a = 9x² -> a = 3
-2a+b = 0

3) a = 3, so b =
2.3 + b =0
6=b

27
Q

With f (x-a) :

If f(x)=2x² +3 and  f(x-a)=2x²-8x+11
  what is the value of a?
A

1) Replace the “x” in f(x) by (x-a) :
2 (x-a)² + 3 = 2 ( x² -2ax + a² ) + 3 =>
2x² - 4ax + 2a² + 3

2) COMPARE with the f(x-a)
2x² - 4ax + 2a² + 3
2x²-8x+11

-4ax = -8x -> a = 2
2a² + 3 = 11

28
Q

With g(x+a):

If ( g(x) ) = 1 / (x-2) (x+2) and g(x+a)= 1/ (x+2 ) (x+6) ;
a =

A

1) Replace x with x+a
1/ (x+a - 2 ) ( x+a + 2)

2) COMPARE
1/ (x+a - 2 ) ( x+a + 2) = 1/ (x+2 ) (x+6)
(x+a - 2 ) ( x+a + 2) = (x+2 ) (x+6)

a - 2 = 2 = 4
a + 2 = 6 = 4
a=4

29
Q

h(x) = x+2/ x - 5 ; what equation is equivalent to h (h(x))

A

h(x) inside h(x) = replace h(x) in x value of h(x)
(x+2/ x - 5) + 2 / (x+2 / x - 5 ) - 5

2) MMC : x+2/ x - 5 + 2 and x+2/ x - 5 - 5
x+2 + 2 (x-5) = x+2 + 2x - 10 = 3x - 8
x+2+ -5 ( x-5) = x+2 - 5x + 25 = -4x +27
3) Go back to ratio form
3x-8/ -4x+27

30
Q

Scatterplot graphs:

y = ax + b

Explain how to discover each letter

A

1) Draw a line from the most-left to the most right

a = slope : ( 0,y1 ) (x2,y2) = y2 - y1/ x2- 0
Isn’t obligatory to x=0. Wherever the line exactly pass through the points is valid

b = The first y-value that the line begins
or replace the values found in the a ( x1,y1) or (x2,y2) and replace in the equation to discover the b

! When the line in the graph is increasing => slope is + and if is exponential, the base is > 1 . EX: (1.05)^x
And vice-versa.

31
Q

If 2+2(8347-4783) = -2j, how much is - j ?

A

SIMPLIFY the 2s
1 + 8347-4783 = -j
1 + 3564 = 3565 = -j

32
Q

(3-4t) - 12 /2 < 2t + 3/2

(3-4t) - 12 is the numerator of 2
only 3 is the numerator of 2

A

Do not distribute “-“ to (3-4t)
THEY ARE SUBCTRATING :
-9 - 4t/2 < 2t + 3/2

Take out the denominator, by multiplying both sides 
-9 - 4t < 2.(2t +  3/2 )
-9 - 4t < 4t + 3 
8t < 12 
t < 3/2
33
Q

WHAT IS THE VALUE OF B IF THERE ARE NO SOLUTIONS
“ “ “ “ INFINITIVE SOLUTIONS

10+7x=4x+2+bx
In the equation shown above, b is a constant. For what value of bbb does the equation have no solutions?

3+11x=cx+2(4x−1)+5

A

NO SOLUTIONS : The normal values should NOT be equal , therefore B = Ax
INFINITIVE SOLUTIONS: The normal values SHOULD BE equal

10+7x=4x+2+bx : 10+ 7x-4x = 2 + bx => 10 + 3x = 2 + bx ;
Since 10 is different 2 , bx = 3x = b = 3

3+11x=cx+2(4x−1)+5 : 11x - 8x = -2+ 5 + cx ;
Since 3 = 3 : 3x = cx = 3

34
Q

4(80+n)=(3k)n

In the equation above, k is a constant. For what value of kkk are there no solutions to the equation?

A

NO SOLUTIONS = k should be equal to A.n
4n = (3k)n
4/3 = k -> Therefore, k.3 = 4

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