
Solution
Given expression
(a^{3}  1) / (b  1)
Substitute a by 2 and b by 2 in given expression
((2)^{3}  1) / ((2)  1)
Simplify expression above
(8  1) / (2  1) = 3

Solution
Given equation
 2 (x + 9) = 20
Solve for x
 2 x  18 = 20
 2 x = 38
Multiply both sides of the above equation
2( 2 x) = 2( 38)
Simplify to obtain
 4 x = 76

Solution
x is a variable
20% of x is written as
20% x
the fifth of x is written as
(1 / 5) x
"20% of it is added to its fifth" is written as:
20% x + (1 / 5) x
seven tenths of x is written as
(7 / 10) x = 0.7 x
the result is equal to 12 subtracted from seven tenths of x
20% x + (1 / 5) x = 0.7x  12
Change % and fractions to decimal numbers
0.2 x + 0.2 x = 0.7x  12
Solve for x
12 = 0.3 x
x = 40

Solution
Given formula
A = 0.5 (b + B) h
Divide both sides of the formula by 0.5 h
(A / (0.5 h)) = 0.5 (b + B) h / (0.5 h)
simplify
(A / (0.5 h)) = b + B
Solve for B
B = (A / 0.5 h)  b
Note that 1 / 0.5 = 2, hence
B = 2 A / h  b

Solution
Let x be the amount of money in dollars that Tom has
Alex has the third of Tom
(1 / 3) x
Linda has twice as much as Alex
2 (1 / 3) x
All together they have 120 dollars, hence
x + x / 3 + 2 x / 3 = 120
Multiply all terms of the above equation by 3, simplify and solve for x
3 x + x + 2 x = 360
x = 60
Linda has
2 (1 / 3) x = (2 / 3) 60 = $40

Solution
Given
6 x^{2}  11 x  2
The linear terms that will make the factors has coefficients 6 and 1 (since 6*1 = 6) or 2 and 3 (since 2*3=6)
(6x )(x )
(2x )(3x )
The constant terms should 1 and 2 or 1 and 2
(6 x  1)(x + 2)
If (6x  1)(x + 2) is expanded, it does not give 6 x^{2}  11 x  2.
(6 x  1)(x + 2) = 6 x^{2 } + 11 x  2
We now try (6x + 1)(x  2) which when expanded gives 6x^{2 }  11 x  2
(6 x + 1)(x  2) = 6 x^{2 }  11 x  2

Solution
Given
x^{2}  7 x  8
Factor
(x + 1)(x  8)
One of the listed factors is
x + 1

Solution
Given
(2xy^{2}  3x^{2}y)  (2x^{2}y^{2}  4x^{2}y)
Eliminate brackets taking signs into consideration
= 2xy^{2}  3x^{2}y  2x^{2}y^{2} + 4x^{2}y
Group like terms  3x^{2}y and 4x^{2}y
= 2xy^{2}  2x^{2}y^{2} + x^{2}y

Solution
Average speed is given by
Total distance / total time = 70 miles / hour
Total distance is given by
x + 200 miles
Total time is equal to
2 + 3 = 5 hours
Substitute in formula above
(x + 200) / 5 = 70
Solve above equation for x
x + 200 = 350
x = 150 miles

Solution
Rewrite all 4 equations in slope intercept form
(I) y = ( 3 / 2) x + 3 / 2
(II) y = (2 / 3) x  5 / 3
(III) y = (2 / 3) x  3 / 2
(IV) y =  3x + 9 / 2
The slopes of all 4 lines are
(I) 3/2
(II) 2/3
(III) 2/3
(IV) 3
For two lines with slopes m and n to be perpendicular, the product of their slopes must equal 1 or m*n = 1.
The product of the slopes of equations (I) and (III) is given by
(3 / 2)*(2 / 3) =  1
hence (I) and (III) are perpendicular.

Solution
Given
f (x) = (x + 1)^{2}
To find f (t + 2), we substitute x by t + 2 in f (x); hence
f (t + 2) = ((t + 2)+ 1)^{ 2}
then simplify and expand
= (t + 3)^{ 2}
= t ^{2 }+ 6t + 9

(√x + √y) (√x  √y)  (√x  √y)^{2} =
Solution
Given
(√x + √y) (√x  √y)  (√x  √y)^{ 2}
Expand (√x + √y) (√x  √y)
(√x + √y) (√x  √y) = (√x)^{ 2}  (√y)^{ 2} = x  y
Expand (√x  √y)^{2}
(√x  √y)^{2} = (√x)^{ 2} + (√y)^{ 2}  2√x √y = x + y  2√x √y
Hence
(√x + √y) (√x  √y)  (√x  √y)^{ 2}
= (x  y)  (x + y  2√x √y)
=  2 y + 2√x √y

x / 2  y / 4 = 7
Solution
Equation of line given in standard form
x / 2  y / 4 = 7
To determine the slope, we need to write the given equation in slope intercept form. Multiply all terms by 4 and simplify and solve for y.
2 x  y = 28
y = 2x  28
Slope is the coefficient of x and is equal to
2

(x / (x  3) + 1 / 2)(2/(x  1)) =
Solution
Given
(x / (x  3) + 1 / 2) (2 / (x  1))
We first simplify the expression ( x / (x  3) + 1 / 2). Reduce to the lowest common denominator.
( x / (x  3) + 1 / 2)
2x / ( 2 (x  3) ) + ( x 3) / (2 (x  3))
= (3x  3) / (2(x  3))
Hence
x / (x  3) + 1 / 2) (2 / (x  1))= (3x  3) (2/(x  1))) / (2(x  3))
= 3(x  1)(2 / (x  1)) / (2(x 3))
Simplify
= 3 / (x  3)

Solution
Let a and b be the x and y coordinates of point B, hence
M is the midpoint of A and B
Since the midpoint M(3,2) is known and point A(2,1) is also known, we can write
(3,2) = ( (2 + a)/2 , (1 + b)/2 )
Two equations may be written as follows
3 = (2 + a) / 2 and 2 = (1 + b) / 2
Solve for a and b
a = 4 and b = 3