Algebra Formulas

The following table contains alternative ways of expressing algebraic functions.

General

a-x = 1 / ax

a1/x = x√a

a-1/x = 1 / x√a

(a.b)x = ax . bx

(ax)y = ax.y

aˣ⁺ʸ = aˣ . aʸ
aˣ⁻ʸ = aˣ / aʸ

e.g.: a2.5 = a² . a0.5
e.g.: a0.25 = a¹ / a0.75

xⁿ/yⁿ + 1 = (xⁿ + yⁿ) / yⁿ

(x - a) / A = x/A - a/A

(x² - a²) = (x - a).(x + a)

Simplify: (x + b)/(x² - a²)
Note: (x² - a²) = (x - a).(x + a)

(x + b)/(x² - a²) = (x + b) / [(x - a).(x + a)]
(x + b) / [(x - a).(x + a)] = A / (x - a) + B / (x + a) {where A & B are unknown}
(x + b) = A.[(x - a).(x + a)] / (x - a) + B.[(x - a).(x + a)] / (x + a)
(x + b) = A.(x + a) + B.(x - a)
Find A:
set x = a
(x + b) = A.(a + a) + B.(a - a)
(x + b) = 2.a.A
A = (x + b) / 2.a
Find B:
set x = -a
(x + b) = A.(-a + a) + B.(-a - a)
(x + b) = B.(-a - a)
B = (x + b) / 2.-a
(x + b)/(x² - a²) = (x + b)/[2.a.(x - a)] + (x + b)/[2.-a.(x + a)]
cancel (x + b):
1/(x² - a²) = 1/[2.a.(x - a)] - 1/[2.a.(x + a)]

Continue in order to prove the above:
1/[(x - a).(x + a)] = 1/[2.a.(x - a)] - 1/[2.a.(x + a)]
1 = [(x - a).(x + a)]/[2.a.(x - a)] - [(x - a).(x + a)]/[2.a.(x + a)]
1 = (x + a)/(2.a) - (x - a)/(2.a)
1 = [1/(2.a)] . [(x + a) - (x - a)]
2.a = [(x + a) - (x - a)]
2.a = x + a - x + a
2.a = a + a

Factorial: e.g. 5! = 5x4x3x2x1

Binomial

(a+b)n = an + nC1.a(n-1).b + nC2.a(n-2).b2 + ..... + nCr.a(n-r).br + ..... bn
Where:
nCr = n! / (n-1)!.r!
nC1 = n
nC2 = n! / (n-2)!.2! = n.(n-1) / 2!
nC3 = n! / (n-3)!.3! = n.(n-1)(n-2) / 3!
etc.
first few terms are as follows;
(a+b)n = an + n.a(n-1).b + n(n-1).a(n-2).b²/2! + n(n-1)(n-2).a(n-3).b³/3! + ..... + bn

If; 0 = ax² + bx + c
then; x = -b ± (b² - 4.a.c)½ / 2.a

Simple and Compound Interest

The following table contains formulas for calculating simple and compound interest.

Where: P = the principal sum, p = percentage interest, n = payment term (years), q = payments per year, I = interest paid over full term, m = amount of each payment & Pn = total amount paid over 'n' years

Simple

I = P.p.n

Pn = I + P

Compound

I = Pn - P

Pn = P.(1 + p/q)n.q

m = Pn / n.q

Net-Present-Value

V = Pn / (1+p/q)n.q

Discount(simple) = Pn(simple) - V

Discount(compound) = Pn(compound) - V

Progressions

The following table contains the formulas for arithmetic and geometric progressions.

Note: r = 2nd term ÷ 1st term, d = 2nd term - 1st term, n = number of terms

Arithmetic

nth term = 1st term + d.(n-1)

Σn terms = n.[2 . 1st term.(n-1)] / 2

Geometric

nth term = 1st term . r(n-1))

Σn terms = 1st term . (1 - rn) / (1 - r)      [r < 1]

Σn terms = 1st term . (rn - 1) / (r - 1)      [r > 1]

Logarithms

The following table contains alternative ways of expressing logarithmic functions.

Note: 'base' refers to the logarithmic base, which can be any positive number
The most common bases are 10 and 2.71828182845905 (the base for natural logs, normally written thus 'ln(x)')

e = 1 + 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ..... = 2.71828182845905

ex = 1 + x¹/1 + x²/2 + x³/3 + x⁴/4 + x⁵/5 + x⁶/6 + x⁷/7 + ....

e-x = 1 - x¹/1 + x²/2 - x³/3 + x⁴/4 - x⁵/5 + x⁶/6 - x⁷/7 + ....

y = Logbase(basey)

Log x = Logbase(x)
Anti-Log x = basex

logbase(x) = loga(x) / loga(base)
Where 'a' can be any number; 1, 2.71828182845905, 10, etc.

base(a+b+c) = base(a).base(b).base(c)
e.g.: exp(a+b+c) = exp(a).exp(b).exp(c)

Logbase(x) - Logbase(y) = log10(base).x/y

Logbase(xa) = a.Logbase(x)

Logbase(x) - Logbase(y) = z
Same as: x/y = basez

1/exp(x) = exp(-x)

eln(x) = x

Xn = Y
n = Log(Y) / Log(X)

Simultaneous Equations

I'll begin by giving you the answer for a 3ʳᵈ-order series, and then show you how to reach it by calculation.

If we set x=1.5, y=-0.5 & z=0.3
the following third-order set of simultaneous equations will generate these results:
(a) 6x - 4y + 3z = 11.9
(b) 2x - 2y + 9z = 6.7
(c) 5x + 8y - 2z = 2.9

factor1 = 3/9 = 0.33333333 [(a)z ÷ (b)z]
multiply each value in (b) by factor1 to create (d):
(d) 0.666666667x - 0.666666667y + 3z = 2.233333333

subtract (a) from (d):
(e) -5.333333333x + 3.333333333y + 0z = -9.666666667

factor2 = 2/5 = 0.4 [(b)x ÷ (c)x]
multiply each value in (c) by factor2 to create (f):
(f) 2x + 3.2y - 0.8z = 1.16

subtract (b) from (f):
(g) 0x + 5.2y - 9.8z = -5.54

factor3 = -4/8 = -0.5 [(a)y ÷ (c)y]
multiply each value in (c) by factor3 to create (h):
(h) -2.5x - 4y + 1z = -1.45

subtract (a) from (h):
(i) -8.5x + 0y - 2z = -13.35

factor4 = -9.8/-2 = -0.5 [(i)z ÷ (g)z]
multiply each value in (i) by factor4 to create (j):
(j) -41.65x + 0y - 9.8z = -65.415

subtract (g) from (j):
(k) -41.65x - 5.2y + 0z = -59.875

factor5 = -5.333333333/-41.65 = 0.12805122 [(k)x ÷ (e)x]
multiply each value in (k) by factor5 to create (l):
(l) -5.333333333x - 0.665866347y + 0z = -7.667066827

subtract (e) from (l):
(m) 0x - 3.99919968y + 0z = 1.99959984

x = (-9.666666667 - 3.333333333*-0.5) / -5.33333333 = 1.5 [from (e)]
y = 1.99959984 / -3.99919968 = -0.5 [from (m)]
z = (-5.54 - 5.2*-0.5) / -9.8 = 0.3 [from (g)]

The above is deliberately long-winded to accentuate the things you can do to obtain the correct result. You may well find a shorter procedure, but it should follow the same rules.
A 2ⁿᵈ-order sequence will be faster, and a 4ᵗʰ-order sequence (and greater) will take much longer, but they can all be resolved using exactly the same procedure; result by elimination.

Determinants

Determinants are a means of solving simultaneous equations, e.g.
a₁.x + b₁.y + c₁.z = 0
a₂.x + b₂.y + c₂.z = 0
a₃.x + b₃.y + c₃.z = 0

which can be written thus:
|a₁,b₁,c₁|
|a₂,b₂,c₂|
|a₃,b₃,c₃|

The following table contains the procedure for solving determinants.

2ᴺᴰ Order

a₁.w + b₁.x = 0
a₂.w + b₂.x = 0

|a₁,b₁| = a₁.b₂ - a₂.b₁
|a₂,b₂|

To solve a 2ᴺᴰ Order equation you perform the calculation as shown above

3ᴿᴰ Order

a₁.w + b₁.x + c₁.y = 0
a₂.w + b₂.x + c₂.y = 0
a₃.w + b₃.x + c₃.y = 0

|a₁,b₁,c₁| = a₁.|b₂.c₂| - a₂.|b₁.c₁| + a₃.|b₁.c₁|
|a₂,b₂,c₂|         |b₃.c₃|       |b₃.c₃|         |b₂.c₂|
|a₃,b₃,c₃|

To solve a 3ᴿᴰ Order equation you convert to 2ᴺᴰ Order equations as shown above then solve 2ᴺᴰ Order equations

4ᵀᴴ Order

a₁.w + b₁.x + c₁.y + d₁.z = 0
a₂.w + b₂.x + c₂.y + d₂.z = 0
a₃.w + b₃.x + c₃.y + d₃.z = 0
a₄.w + b₄.x + c₄.y + d₄.z = 0

|a₁,b₁,c₁.d₁| = a₁.|b₂.c₂.d₂| - a₂.|b₁.c₁.d₁| + a₃.|b₁.c₁.d₁| - a₄.|b₁.c₁.d₁|
|a₂,b₂,c₂,d₂|        |b₃.c₃.d₃|       |b₃.c₃.d₃|         |b₂.c₂.d₂|        |b₂.c₂.d₂|
|a₃,b₃,c₃,d₃|        |b₄.c₄.d₄|       |b₄.c₄.d₄|         |b₄.c₄.d₄|        |b₃.c₃.d₃|
|a₄,b₄,c₄,d₄|

To solve a 4ᵀᴴ Order equation you convert to 3ᴿᴰ Order equations as shown above, then convert to 2ᴺᴰ Order equations then solve 2ᴺᴰ Order equations

The same procedure may be followed for all subsequent Order equations: 5ᵀᴴ, 6ᵀᴴ, 7ᵀᴴ, etc.

Graph with a positive slope
Fig 1. Finding x and y

Slope

A calculation method to find a point on a graph of positive or negative slope

Positive Slope

with reference to Fig 1

x = (x₂-x₁).(y-y₁)/(y₂-y₁) + x₁

y = (y₂-y₁).(x-x₁)/(x₂-x₁) + y₁

 

Negative Slope

Graph with a negative slope
Fig 2. Finding x and y

with reference to Fig 2

x = (x₂-x₁).(y-y₂)/(y₁-y₂) + x₁

y = (y₁-y₂).(x-x₁)/(x₂-x₁) + y₂

 

Geometry

Properties of a Sphere

Circumference: C = 2.π.r
Surface Area: A = 4.π.r²
Volume: V = ⁴/₃.π.r³
Distance between 'n' equally spaced points on the surface:
Arc: d = π.A / C.n = 6.π.V / A.n
Linear: ℓ = 2.r.Sin(½.d/r)
to find 'n':
n = π/Asin(½.ℓ/r)

If ℓ = r then:
n = π/Asin(½) = 6
In this special case; 'n = 6' is a constant, irrespective of the spherical radius

 

Further Reading

You will find further reading on this subject in reference publications(19)

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