3cddb93a-deb0-4480-ade5-b645fc11bf90 54 15 12 decimal

&[DATE] &[TIME] - &[FILENAME]

Circle/line intersection

3 appVersion 0.98.6179 ≡

C 6 : Circle ≡

L 0 : Line ≡

Φ vector equalities Φ x 2^y 2^+ C - 0 ≡ x y + L - 0 ≡ 21 mat :

init x 1.5 ≡ y 1.5 - ≡ 21 mat :

TOL 1013 -^:

Does not support assignment Φ init TOL FindRoot 1.732051 1.732051 - 21 mat

Assign the solutions sol x y 21 mat : 1.732051 1.732051 - 21 mat

C x 2 ^ - sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 2 5 mat 3 1 sys

assign the solutons [x,y] ... for sanity x y find Φ init TOL FindRoot : 1.7321 1.7321 - 21 mat

sanity x 2^y 2^+ x y + 21 mat 6.000000000000020 0.000000000000000 21 mat

x y Clear 1

Solving the ellipse/line intersection [Smath Designers]

============================================== solving 1rst intersection

determined condition
determined condition f 2 x 2^* 3 y 2^* + 59 - 0 ≡ x - 4 y * + 8 - 0 ≡ 21 mat :

Guess roughly x.0 x 1.25 ≡ y 0.75 ≡ 21 mat :

Accuracy TOL 1015 -^:

most robust for this example sol_1 f x.0 TOL LevenbergMarquardt :

sol_1 4321 mat

Ellipse in context x y E 2 x 2^* 3 y 2^* + 59 - :

u λine u 8 + 4 / :

x y axes x y / - :

============================================== solving 2nd intersection

x y Clear 1

determined condition
determined condition Φ 2 x 2^* 3 y 2^* + 59 - 0 ≡ x - 4 y * + 8 - 0 ≡ 21 mat :

Guess wisely y.0 x 7 - ≡ y 0.75 ≡ 21 mat :

Accuracy TOL 1015 -^:

most robust for this example sol_2 Φ y.0 TOL LevenbergMarquardt :

sol_2 5.371429 - 0.657143 21 mat

E u λine sol_1 1 el sol_1 2 el 1 2 mat axes sol_2 1 el sol_2 2 el 1 2 mat 5 1 sys

E u λine sol_1 1 el sol_1 2 el 12 mat axes sol_2 1 el sol_2 2 el 12 mat 51 sys

r 0.5 - : x y q x 2^y 2^+ r + : x y axes x y / - : 31 sys

solving  "one" pair u 0.5 :

x circle x y q y solve :

pair wise X u circle : 0.5 - 0.5 21 mat

xy u X 2 el augment :

q xy axes u 0.75 u 0.75 - 2 2 mat 4 1 sys

r 0.25 - : x y c x 2^y 2^+ r + : 21 sys

Collect pairwise

solving  "ω" pairs ω 0.31 0.45 0.315 range :

x circle collect x y c 0 ≡ y solve : collect 21 line :

pairs ω ω circle vectorize augment :

pairs 0.31 0.3923 - 0.3923 21 mat 0.315 0.3883 - 0.3883 21 mat 0.32 0.3842 - 0.3842 21 mat 32 mat

rejects  range that
create '0' circle. 0.3 0.45 0.31 range

Collect augmented

solving /augment "ω" pairs ω 0.25 0.45 0.251 range :

x y Clear r 0.25 - : x y c x 2^y 2^+ r + : 31 sys

x circle collect x y c y roots : collect 21 line :

pair wise pairs ω circle vectorize : 0.433013 0.432434 0.431852 0.431267 0.430679 51 mat

Collect ω ω circle vectorize ω circle vectorize - augment :

Collect 0.25 0.433 0.433 - 0.251 0.4324 0.4324 - 0.252 0.4319 0.4319 - 0.253 0.4313 0.4313 - 0.254 0.4307 0.4307 - 0.255 0.4301 0.4301 - 0.256 0.4295 0.4295 - 73 mat

cercle a 1 : b 1 : xo 1 : X 1 : Y 1 : 15 mat

ellipse a 1 : b 1.5 - : xo 1 : X 1 : Y 1 : 15 mat

strophoid a 1 : b 1.5 : xo 1 : X 0.75 : Y 1 : 15 mat

strophoid x f 1 b / 1 x xo - (2^a 4^/ - sqrt x xo - (/ * a 4^* x X - (* Y + :

x q x :

true
extraterrestrial
true sol x f x q - x 0.75 1.75 solve : 0.807 1.6746 21 mat

x f x f - 2 + a 5 a 10 - 2 2 mat 1 x q 0.807 0.807 f . 14 black 1 5 mat 1.6746 1.6746 f . 14 black 1 5 mat 7 1 sys

x f x f - 2 + a 5 a 10 - 22 mat 1 x q 0.807 0.807 f . 14 black 15 mat 1.6746 1.6746 f . 14 black 15 mat 71 sys

Strophoid

a π - :

xx q 1 xx * 0.5 + :

u v Strophoid u u 2^v 2^+ (* a a cos u 2^v 2^- (* * + 2 a sin * u * v * + :

Strophoid xx q 2 1 sys

x f x y Strophoid 0 ≡ y solve maple :

x f x π - (x π + (* - sqrt x * x π - / x π - (x π + (* - sqrt x * x π - / - 21 mat

x π - ( x π + ( * - sqrt x * x π - / ± π - 2 π - 2 - 2 2 mat x 3 1 sys

x f x π - (x π + (* - sqrt x * x π - / x π - (x π + (* - sqrt x * x π - / - 21 mat :

x π - :

x f 0.000000000000000 0.000000000000000 21 mat

xx f 1 el xx q - 0 ≡ xx 1 - 0 solve xx f 2 el xx q - 0 ≡ xx 2 - 2 solve 21 mat 0.260378 - 1.353725 - 1.114103 21 mat 21 mat

Solving Ellipse otherwise

NOT needed on new document x y Clear 1

determined condition
determined condition system 2 x 2^* 3 y 2^* + 59 - 0 ≡ x - 4 y * + 8 - 0 ≡ 21 mat :

Accuracy TOL 109 -^:

Guess roughly init x 1.75 ≡ y 0.75 ≡ 21 mat :

sol_1 system init TOL FindRoot : 4.000000000000000 3.000000000000000 21 mat

clear from previous use x y Clear 1

Guess roughly init x 7 - ≡ y 0.75 ≡ 21 mat :

sol_2 system init TOL FindRoot : 5.371428571428890 - 0.657142857142777 21 mat

Ellipse in context x y E 2 x 2^* 3 y 2^* + 59 - :

xx λine xx 8 + 4 / :

x y axes x y / - :

E xx λine sol_1 1 el sol_1 2 el 1 2 mat axes sol_2 1 el sol_2 2 el 1 2 mat 5 1 sys

E xx λine sol_1 1 el sol_1 2 el 12 mat axes sol_2 1 el sol_2 2 el 12 mat 51 sys

simply on new document x λine x 8 + 4 / :

The discrete iterative solve bloc

NASA(h) is kPa vs h[m] ... we solve [m] actual altitude vs measured [kPa] h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

60000 NASA 25000 NASA 11000 NASA 0 NASA 41 mat 0.0279 2.5223 22.7074 101.4009 41 mat

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L 60000 NASA : H 101.4009 : N 200 : 13 mat u L H N xd : 21 sys

altitude i 1 u rows range altitude i el h NASA u i el - 0 ≡ h 0100000 solve : for u 4 round vectorize altitude 0 round vectorize augment 21 line :

t0 1 time :

NASA(h) solved [m_altitude] from measured kPa

altitude 101.325 0 + 40 red 1 5 mat 11000 NASA 11000 + 100 red 1 5 mat 25000 NASA 25000 + 175 red 1 5 mat 4 1 sys

1 time t0 - min 1.4

MACH on canvas

γ 1.4 : (Air isentropic ≡ k 0.88128485 : (Pitot constant ≡ p static pressure ≡ qc dynamic pressure ≡ x Pitot dynamic ratio qc/P ≡ x qc p / ≡ (supersonic Pitot ratio ≡ 61 line

Observe the canvas solutions:

as 'x' in Supersonic(x) runs on the plot canvas,

the 'roots' solves and plots each canvas pixel result ...

each result on the basis of 96 ppi [pixel per inch].

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Plot limit x c 10 x ≤ (x 5 ≤ (& cases :

t0 1 time :

x Subsonic 2 γ 1 - / (x 1 + (γ 1 - γ /^1 - (* sqrt 0 x ≤ (x k ≤ (& Supersonic cases : x Supersonic U x 1 + (117 U 2^* / - (2.5^* sqrt / k - 0 ≡ U 1 roots k x ≤ Subsonic cases : 21 line

x Subsonic x Supersonic x c * k 1 + 20 black 1 5 mat 2 0 Pitot-Mach 12 red 1 5 mat 4 1 sys

1 time t0 - 71 s *

supersonic

subsonic

Otherwise profitable Discrete "roots" at(2) . . .

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x y u Clear 1

r 0.25 - : x y c x 2^y 2^+ r + : 21 sys

lo 0.0001 : hi 0.4999 : N 99 : 13 mat

find low limit u lo hi lo hi lo - N / + (range :

t0 1 time :

C j 1 u length range rho j el ρ u j el y c y roots at : 11 line for rho 21 line :

u C augment u C - augment u - C augment u - C - augment 4 1 sys

1 time t0 - 9.5 s *

x y u Clear 1

x y Φ3 x 2^y 2^+ (2^1 x 3^3 x * y 2^* - (* - : x C 1 x 2^- sqrt ± : 21 sys

x map x y Φ3 0 ≡ y solve maple :

junction 0.5625 - 0.5625 - map 1 el + 15 black 15 mat :

plot x map 1 el x map 2 el x map 3 el x map 4 el 41 sys :

plot x C junction 3 1 sys

Maple  symbolic x q1 x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / : x q2 x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / - : 21 sys

lo 0.56245 - : hi 0.99 : N 99 : 13 mat

Hi 0.001 - :

find low limit u lo hi lo hi lo - N / + (range :

find low limit v lo Hi lo Hi lo - 51 / + (range :

t0 1 time :

T j 1 u length range rho j el dum u j el y Φ3 y roots at : 11 line for rho 21 line :

u T augment u T - augment v v q1 vectorize augment v v q2 vectorize augment 4 1 sys

1 time t0 - 27 s *

u T augment u T - augment v v q1 vectorize augment v v q2 vectorize augment 41 sys

Trifolium u T augment u T - augment v v q1 vectorize augment v v q2 vectorize augment stack :

Trifolium

Otherwise profitable Discrete-Difference "roots" at(3) . . .

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r 0.125 - :

x y u Clear 1

x y Γ x 2^y 2^+ r + :

x y Ω 1 x 4^(* 2 x 2^* y * + 3 x * y 2^* - 1 y 3^(* + :

Ω Γ 0.35355 0.04526 - 1 2 mat 3 1 sys

Intersection limit

t0 1 time :

lo 0.35 : hi 0.36 : N 1000 : 13 mat

find low limit u lo hi lo hi lo - N / + (range :

last value "on error" ... NO real root. data j 1 u length range rho j el at3 u j el y Ω y roots u j el y Γ y roots at : break try for rho 21 line :

1 time t0 - 0.8 s *

data data :

"real root" exists in the positive region left of 0.35355. On the other side, the differential region is negative ... thus => "NO real root".

k 1 data rows range : junction u k el data augment : junction junction : 31 sys

u 0.35355 : Yjunction u y Γ u y Ω ≡ y roots : 21 sys

1 time t0 - min 0.7

Yjunction 0.04526 -

junction junction rows row 0.35355 0.0525 12 mat

junction 0.35355 0.5 0.35355 0.05 - 2 2 mat 0.35 0.0525 0.36 0.0525 2 2 mat 3 1 sys

real root

NO real root at(3)

x y Clear 1

a 816 π * sqrt * : 351.67021

x y gn 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.0025 - :

About the Electron function . . .

A closed lobe that has no symbolic expression

of the ± branches. It complicates more familiar

expansion of f(x,y) but delights Smath.

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x y rd 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.005 + :

x y axes x y / - :

solving 6 values u 7.5 :

====================================================

axes rd gn u 25 u 25 - 2 2 mat 14.2455293 5 14.2455293 5 - 2 2 mat 5 1 sys

x green user x y gn y solve : x user 12 mat 21 line :

axes rd gn u 25 u 25 - 22 mat 14.24553 5 14.24553 5 - 22 mat 51 sys

pair wise u green 7.5 18.063615 - 6.294379 - 6.294379 18.063615 41 mat 12 mat

x red user x y rd y solve : x user 12 mat 21 line :

pair wise u red 7.5 3.716206 - 3.716206 21 mat 12 mat

x f 5953 / 11 x 2 / - (2^- (* :

Solving a function, i.e: inverse

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1 f 3 f 2 f 1.75 f 41 mat 148.75 148.75 198.333 195.234 41 mat

x f 0 ≡ x 35 solve 4

x c 10 x ≤ (x 4 ≤ (& cases :

algo style "iterative inverser" on range [L, H, N] .... f L H N Inverse dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x roots : for ω sol augment 41 line :

Inv f 0 198.333 100 Inverse :

x freak Inv 1 col Inv 2 col x linterp :

ξ 0.75 :

ξ f ξ f freak 21 mat 120.859 0.75 21 mat

ξ 148.75 :

=============================================

x f x c * 0 ξ ξ freak ξ ξ freak 0 3 2 mat 2 256 2 64 - 2 2 mat 3 1 sys

50 freak 148.75 freak 195.234 freak 198.333 freak 41 mat 0.27 1 1.753 1.883 41 mat

ξ freak 1

2. Maple classical/direct analytical inverser

x a b q 1 x b / - exp (a^- :

y x a b q ≡ x solve maple Pile vide.

y a b sol 1 y - ln b * a / - :

a 2 : b 1 : 12 mat

x middle b x * :

x a b q x a b sol x middle 3 1 sys

An inverse function:

1. may exist simply, this example: q(x,a,b)

2. may exist via "horrible maths" !!!.

3. may not exist at all ... inverse ∫(x)

4. inverse ∫(x) is evaluated via the iterative "solve bloc".

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x a b q x a b sol - 0 ≡ x 0.5 1 solve 0.797

4. Golden-Ratio

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1 ≥ y2 ≡ "max in range" y1 ≤ y 2≡ "min in range"

x f x 4^2 x 3^* - 7 x 2^* - 2 - : a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : 21 line

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema y1 y2 ≥ max in range ≡ y1 y2 ≤ min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 61 line :

TOL 1012 -^:

X f 1.5 - 1 - y1 y2 ≤ TOL Extrema 1 el f 0.5 - 0.5 y1 y2 ≥ TOL Extrema 1 el f 2.5 3 y1 y2 ≤ TOL Extrema 1 el 31 mat :

x f X 1 el X 1 el f + 15 red X 2 el X 2 el f + 15 red X 3 el X 3 el f + 15 red 3 5 mat 2 1 sys

X 1.265564 - 0 2.765564 31 mat

iter f 1.5 - 1 - y1 y2 ≤ TOL Extrema 2 el f 0.5 - 0.5 y1 y2 ≥ TOL Extrema 2 el f 2.5 3 y1 y2 ≤ TOL Extrema 2 el 31 mat :

iter 56585631 mat

x f x 4^2 x 3^* - 7 x 2^* - 2 - : x g x : 21 line

solve root(s) X x f x 3 - 5 solve : 1.9225 - 3.8522 21 mat

solve intersection(s) C x f x g - 0 ≡ x 3 - 5 solve : 1.837 - 3.8963 21 mat

plot x f x g X 1 el X 1 el f . 15 red 15 mat X 2 el X 2 el f . 15 red 15 mat C 1 el C 1 el f . 15 black 15 mat C 2 el C 2 el f . 15 bkack 15 mat 61 sys :

plot

x y Clear 1

Solving numeric a user system ======================

b 1 : c π : 21 line

f a0 0 ≡ a1 c ≡ a1 a2 + a3 + 0 ≡ a1 2 a2 * + 3 a3 * + b ≡ 41 mat :

init a0 1 ≡ a1 1 ≡ a2 1 ≡ a3 1 ≡ 41 mat :

a0 a1 a2 a3 sol f init FindRoot :

solutions in lexicographic order of "init" solutions a0 a1 a2 a3 sol vectorize : 1.8504 1017 -^* 3.1416 7.2832 - 4.1416 41 mat

Solving quaternion ======================================

α 0.5 :

β 1.5 :

p q r Clear 1

The Quaternion system f p q + r - 1012 -^≡ p 2^q 2^+ r 2^+ 1 ≡ β p * α r * + 1012 -^≡ 31 mat :

init p 1012 -^≡ q 1012 -^≡ r 1012 -^≡ 31 mat :

Sanity check vs Mathcad 11 f init FindRoot 0.1961 - 0.7845 0.5883 31 mat

p q r 31 mat 0.1961 - 0.7845 0.5883 31 mat

Quaternion p α - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / α 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / q 12 α 2^* 2 α * β * + 2 β 2^* + sqrt / α β + (* 1 - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / α β + (* r 12 α 2^* 2 α * β * + 2 β 2^* + sqrt / β * 1 - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / β * 33 mat :

Next: evaluate numeric

modulo 2 β 2^* 2 β * α * + 2 α 2^* + sqrt :

α 0.5 :

β 1.5 :

Quaternion 2 col 0.1961 - 0.7845 0.5883 31 mat

V α - modulo / α modulo / α β + (modulo / α β + (- modulo / α modulo / α - modulo / 32 mat :

V 0.1961 - 0.1961 0.7845 0.7845 - 0.1961 0.1961 - 32 mat

Next: stack a 'p' value

p 0.6 :

q Quaternion 2 col p stack :

q 0.1961 - 0.7845 0.5883 0.6 41 mat

Nex: the unit vector

q 0.1961 - 0.7845 0.5883 0.6 41 mat :

q q q norme / : 0.1682 - 0.6727 0.5045 0.5145 41 mat

solutions are in lexicographic order

Solving a chemical balance =======================

k1 0.5 : k2 0.25 : k3 0.1 : 31 line

init γ1 1012 -^≡ γ2 1012 -^≡ 21 mat :

f γ1 γ2 + k1 k2 + k3 + ≡ γ1 γ2 * k1 k2 * k1 k3 * + ≡ 21 mat :

f init FindRoot 0.5 0.35 21 mat

γ1 γ2 21 mat 0.5 0.35 21 mat

Clear "initial" to renew the solver

γ1 γ2 Clear 1

k1 10.052 : k2 14.836 : k3 4.982 : 13 mat

init γ1 1012 -^≡ γ2 1012 -^≡ 21 mat :

f γ1 γ2 + k1 k2 + k3 + ≡ γ1 γ2 * k1 k2 * k1 k3 * + ≡ 21 mat :

f init FindRoot 19.818 10.052 21 mat

γ1 γ2 21 mat 19.818 10.052 21 mat

Implicit "roots" =======================

u x Clear 1

user u e :

user [r]

x f 5 x + x 2^- x 3^+ u 21 mat : r x f 1 el x f 2 el - x 1 - roots : z x f 1 el x roots : 31 sys

r z 21 mat 0.869264218088383 - 1.27816307279814 - 21 mat

r f 1 el r f 2 el - z f 1 el 21 mat 91016 -^* 71014 -^* 21 mat

x f 1 el x f 2 el 2 1 sys

"roots" ... Solving a system

U y 3 ≡ 3 x * y + 5 ≡ 21 mat :

v x y 21 mat :

U v roots 0.666666666666667 321 mat

A 013122 mat :

v 3521 mat :

A 1 -^v * 0.666666666666667 321 mat

"roots" ... Intersection

x f x 2^4 - 2 x * 4 x 2^* - 21 mat :

R x f 1 el x f 2 el - x 1 - roots x f 1 el x f 2 el - x 1 roots 21 mat :

x f 1 el x f 2 el R 1 el R 1 el f 1 el + 15 black R 2 el R 2 el f 1 el + 15 black 2 5 mat 3 1 sys

R 0.716515138991168 - 1.11651513899117 21 mat

x f 1 el x f 2 el R 1 el R 1 el f 1 el + 15 black R 2 el R 2 el f 1 el + 15 black 25 mat 31 sys

'roots init' => high accuracy R 1 el f 1 el R 1 el f 2 el - R 2 el f 1 el R 2 el f 2 el - 21 mat 021014 -^* 21 mat

x f 101 x + (* 2 x ln * exp - :

solve & roots ... wisely

x f x f Re x abs f 3 1 sys x abs f x f

x Φ x abs f :

X x abs f x solve : 10.9161 - 10.9161 21 mat

X Φ vectorize 0.000000000000000 0.000000000000000 21 mat

r x f 0 ≡ x 8 roots : 10.9161

r Φ 0.000000000000000

NO real roots x f 0 ≡ x roots

r x f 0 ≡ x max f(x x < roots :

Solving the cylindrical box

D H Clear 1

I hate FindRoot ! f init TOL GivenFind f init TOL FindRoot :

to be minimized 
for a given volume D H SurfArea 2 π D 2 / (2^* (* π D * H * + :

D H Vol π D 2^4 / * H * :

Accuracy TOL 109 -^:

system H D - 0 ≡ D H Vol 10 ≡ 21 sys :

Does not like D ≡ 0
H ±  > D init D 0.001 ≡ H 1 ≡ 21 mat :

D H 21 mat system init TOL GivenFind :

From educated knowledge, we know that for a given volume, the minmum surface area occurs for D = H

D H 21 mat 2.33508865 2.33508865 21 mat

Solving the Goat constant

I hate FindRoot ! f init GivenFind f init FindRoot :

http://mathworld.wolfram.com/GoatProblem.html

Tieing a goat to a point on the interior of the fence with radius a=1 using a chain of length L=r, consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection ... which cannot be solved exactly, but which has approximate solution: L= 1.15872847

R 1 :

L Clear 1

The solve bloc fails miserably ...  if not closely initialized. Goat 212 / R 2^* 2 L 2 R * / asin * 2 L 2 R * / asin * sin - (* (* L 2^L 2 R * / acos * + π R 2^* 2 / ≡ :

init L 1.1401 ≡ :

L Goat init GivenFind (: 1.15872847

R L Goat 212 / R 2^* 2 L 2 R * / asin * 2 L 2 R * / asin * sin - (* (* L 2^L 2 R * / acos * + :

x c 10 x ≤ (x 2 R * ≤ (& cases :

R x Goat x c * L π R 2 ^ * 2 / + 15 red 1 5 mat 2 1 sys

... a more mathematical competitive solution R L Goat π R 2^* 2 / - 0 ≡ L 1 1.2 solve 1.15872847

The pet shop quiz

How many of each species can you purchase with $ 100 for 100 pets ?

x y z Clear 1

f init GivenFind f init FindRoot :

The classroom
proposal solve x y + z + 100 ≡ 15 x * 1 y * + 0.25 z * + 100 ≡ 21 mat

f x y + z + 100 ≡ 15 x * 1 y * + 0.25 z * + 100 ≡ z y - 15 ≡ 31 mat :

init x 1 ≡ y 1 ≡ z 1 ≡ 31 mat :

x dogs ≡ ($ 15 ≡ y cats ≡ ($ 1 ≡ z mices ≡ ($ 0.25 ≡ 31 line

x y z sol f init GivenFind :

... we can establish the purchase There is a missing condition: with $100 you can purchase 400 mices, 0 cat, 0 dog.. For 'z' mices purchsed minus 'y' cats [because you must purchase cats too] from an unknown set of pets, this trading must return the price of a dog, i.e: $ 15. Therefore => the missing condition is ............ [z-y]=15

x dogs [3]
y cats [41]
z mices [56] x y z sol 3415631 mat

sanity x y + z + 15 x * 1 y * + 0.25 z * + 21 mat 10010021 mat

===========================================================================

a b c Clear 1

F 2 a 2^* a b + (2^≡ a 2^b 2^+ 100 ≡ c a b + (2^25 - sqrt ≡ 31 mat :

init a 1 ≡ b 1 ≡ c 1 ≡ 31 mat :

a b c sol F init GivenFind :

a b c sol 9.3301 3.7924 12.5824 1.3665 i * + 31 mat

Complex solution

===========================================================================

ε 1012 -^:

X Y Z Clear 1

sys 2 X * Y + 52 Z 2^* - (- 0 ≡ Y 3^4 Z * + 4 - 0 ≡ X Y * Z + e Z^- 0 ≡ 31 mat :

v X 1 ≡ Y 1 ≡ Z ε ≡ 31 mat :

sys v GivenFind sys v ε FindRoot :

sys v GivenFind 1.422 0.975 0.768 31 mat

X Y Z 31 mat 1.422 0.975 0.768 31 mat

2 X * Y + 52 Z 2^* - (- Y 3^4 Z * + 4 - X Y * Z + e Z^- 31 mat 21014 -^* 11015 -^* - 51015 -^* - 31 mat

... for each variable, calculations have at least 3 decimal places function(s) precision.