#1: " ----------[ D e r i v e ]---------- " User #2: " ---------- Initialization ---------- " User #3: " ---------- Equations ---------- " User #4: " Manipulate an equation using a natural syntax: " User #5: " (x = 2)/2 + (1 = 1) => x/2 + 1 = 2 " User x = 2 #6: ------- + (1 = 1) User 2 x #7: --- + true = true + 1 Simp(#6) 2 User #8: " Solve various nonlinear equations---this cubic polynomial has all real roots " 3 2 #9: SOLVE(3*x - 18*x + 33*x - 19 = 0, x) User Simp(#9) / / pi \ / 2*pi \ | 2*SQRT(3)*SIN|----| 2*SQRT(3)*SIN|------| #10: | \ 9 / \ 9 / |x = 2 - ---------------------, x = 2 - -----------------------, \ 3 3 / pi \ \ 2*SQRT(3)*COS|----| | \ 18 / | x = --------------------- + 2| 3 / #11: " Some simple seeming problems can have messy answers: "User User #12: " x = { [sqrt(5) - 1]/4 +/- 5^(1/4) sqrt(sqrt(5) + 1)/[2 sqrt(2)] i, " User #13: " - [sqrt(5) + 1]/4 +/- 5^(1/4) sqrt(sqrt(5) - 1)/[2 sqrt(2)] i} " 4 3 2 #14: eqn := x + x + x + x + 1 = 0 User #15: q_ := SOLVE(eqn, x) User #16: q_ User Simp(#16) / SQRT(5) 1 / SQRT(5) 5 \ SQRT(5) #17: |x = --------- - --- + #i*SQRT|--------- + ---|, x = --------- - \ 4 4 \ 8 8 / 4 1 / SQRT(5) 5 \ SQRT(5) 1 --- - #i*SQRT|--------- + ---|, x = - --------- - --- + 4 \ 8 8 / 4 4 / 5 SQRT(5) \ SQRT(5) 1 / 5 #i*SQRT|--- - ---------|, x = - --------- - --- - #i*SQRT|--- \ 8 8 / 4 4 \ 8 SQRT(5) \\ - ---------|| 8 // #18: " Check one of the answers " User x := RHS(q_ ) #19: 1 User 4 3 2 #20: x + x + x + x + 1 = 0 User #21: 0 = 0 Simp(#20) #22: x := User #23: eqn := User User #24: " x = {2^(1/3) +- sqrt(3), +- sqrt(3) - 1/2^(2/3) +- i sqrt(3)/2^(2/3)} " #25: " [Mohamed Omar Rayes] " User 6 4 3 2 #26: SOLVE(x - 9*x - 4*x + 27*x - 36*x - 23 = 0, x) User / 6 4 3 2 \ #27: \x - 9*x - 4*x + 27*x - 36*x = 23/ Simp(#26) User #28: " x = {1, e^(+- 2 pi i/7), e^(+- 4 pi i/7), e^(+- 6 pi i/7)} " 7 #29: SOLVE(x - 1 = 0, x) User Simp(#29) / 4/7 - 4/7 - 5/7 #30: \x = 1, x = (-1) , x = (-1) , x = - (-1) , x = - 5/7 - 1/7 1/7\ (-1) , x = - (-1) , x = - (-1) / User #31: " x = 1 +- sqrt(+-sqrt(+-4 sqrt(3) - 3) - 3)/sqrt(2) [Richard Liska] " User 8 7 6 5 4 3 2 #32: SOLVE(x - 8*x + 34*x - 92*x + 175*x - 236*x + 226*x - 140*x + 46 = 0, x) Simp(#32) / 8 7 6 5 4 3 2 #33: \x - 8*x + 34*x - 92*x + 175*x - 236*x + 226*x - 140*x = \ -46/ User #34: " The following equations have an infinite number of solutions (let n be an " #35: " arbitrary integer): " User User #36: " x = {log(sqrt(z) - 1), log(sqrt(z) + 1) + i pi} [+ n 2 pi i, + n 2 pi i] " 2*x x #37: q_ := #e + 2*#e + 1 = z User #38: SOLVE(q_, x) User #39: [x = LN(SQRT(z) - 1), x = LN(- SQRT(z) - 1)] Simp(#38) User #40: " x = (1 +- sqrt(9 - 8 n pi i))/2. Real solutions correspond to n = 0 => " #41: " x = {-1, 2} " User 2 #42: SOLVE(EXP(2 - x ) = EXP(-x), x) User #43: [x = -1, x = 2, x = inf] Simp(#42) User #44: " x = -W[n](-1) [e.g., -W[0](-1) = 0.31813 - 1.33724 i] where W[n](x) is the " #45: " nth branch of Lambert's W function " User #46: SOLVE(EXP(x) = x, x) User / x \ #47: \#e - x = 0/ Simp(#46) #48: " x = {-1, 1} " User x #49: SOLVE(x = x, x) User / x \ #50: \x - x = 0/ Simp(#49) User #51: " This equation is already factored and so *should* be easy to solve: " User #52: " x = {-1, 2*{+-arcsinh(1) i + n pi}, 3*{pi/6 + n pi/3}} " 2 2 3 #53: q_ := (x + 1)*(SIN(x) + 1) *COS(3*x) = 0 User #54: SOLVE(q_, x) User Simp(#54) / pi pi pi #55: |x = -1, x = ----, x = ----, x = - ----, x = #i*LN(SQRT(2) + 1), \ 2 6 6 x = #i*LN(SQRT(2) - 1), x = pi - #i*LN(SQRT(2) + 1), x = pi - #i*LN(SQRT(2) - 1), x = -pi - #i*LN(SQRT(2) + 1), x = -pi - \ #i*LN(SQRT(2) - 1)| / #56: " x = pi/4 [+ n pi] " User #57: SOLVE(SIN(x) = COS(x), x) User / pi 3*pi 5*pi \ #58: |x = ----, x = - ------, x = ------| Simp(#57) \ 4 4 4 / #59: SOLVE(TAN(x) = 1, x) User / pi 3*pi 5*pi \ #60: |x = ----, x = - ------, x = ------| Simp(#59) \ 4 4 4 / #61: " x = {pi/6, 5 pi/6} [ + n 2 pi, + n 2 pi ] " User / 1 \ #62: SOLVE|SIN(x) = ---, x| User \ 2 / / pi 5*pi 7*pi \ #63: |x = ----, x = ------, x = - ------| Simp(#62) \ 6 6 6 / #64: " x = {0, 0} [+ n pi, + n 2 pi] " User #65: SOLVE(SIN(x) = TAN(x), x) User #66: [x = 0, x = pi, x = -pi, x = 2*pi] Simp(#65) #67: " x = {0, 0, 0} " User #68: SOLVE(ASIN(x) = ATAN(x), x) User #69: [x = 0] Simp(#68) #70: " x = sqrt[(sqrt(5) - 1)/2] " User #71: SOLVE(ACOS(x) = ATAN(x), x) User / / SQRT(5) 1 \\ #72: |x = SQRT|--------- - ---|| Simp(#71) \ \ 2 2 // #73: " x = 2 " User / x - 2 \ SOLVE|------- = 0, x| #74: | 1/3 | User \ x / #75: [x = 2] Simp(#74) #76: " This equation has no solutions " User 2 #77: SOLVE(SQRT(x + 1) = x - 2, x) User #78: [] Simp(#77) #79: " x = 1 " User #80: SOLVE(x + SQRT(x) = 2, x) User #81: [x = 1] Simp(#80) #82: " x = 1/16 " User 1/4 #83: SOLVE(2*SQRT(x) + 3*x - 2 = 0, x) User / 1 \ #84: |x = ----| Simp(#83) \ 16 / User #85: " x = {sqrt[(sqrt(5) - 1)/2], -i sqrt[(sqrt(5) + 1)/2]} " / 1 \ SOLVE|x = --------------, x| #86: | 2 | User \ SQRT(1 + x ) / / 2 \ #87: \x*SQRT(x + 1) = 1/ Simp(#86) User #88: " This problem is from a computational biology talk => 1 - log_2[m (m - 1)] " k #89: SOLVE(COMB(m, 2)*2 = 1, k) User / LN(m - 1) LN(m) \ #90: |k = - ----------- - ------- + 1| Simp(#89) \ LN(2) LN(2) / User #91: " x = log(c/a) / log(b/d) for a, b, c, d != 0 and b, d != 1 [Bill Pletsch] " x x #92: SOLVE(a*b = c*d , x) User / LN(a) - LN(c) \ #93: |x = ---------------| Simp(#92) \ LN(d) - LN(b) / #94: " x = {1, e^4} " User #95: SOLVE(SQRT(LOG(x)) = LOG(SQRT(x)), x) User / 4\ #96: \x = 1, x = #e / Simp(#95) User #97: " Recursive use of inverses, including multiple branches of rational " #98: " fractional powers [Richard Liska] " User #99: " => x = +-(b + sin(1 + cos(1/e^2)))^(3/2) " User 2/3 #100:SOLVE(LOG(ACOS(ASIN(x - b) - 1)) + 2 = 0, x) User #101:[] Simp(#100) User #102:" x = {-0.784966, -0.016291, 0.802557} From Metha Kamminga-van Hulsen, " User #103:" ``Hoisting the Sails and Casting Off with Maple'', _Computer Algebra " User #104:" Nederland Nieuwsbrief_, Number 13, December 1994, ISSN 1380-1260, 27--40. " / x - 5 \ 3 #105:eqn := 5*x + EXP|-------| = 8*x User \ 2 / #106:SOLVE(eqn, x) User / x/2 3 5/2 5/2 \ #107:\#e - 8*x *#e + 5*x*#e = 0/ Simp(#106) #108:Precision := Approximate User #109:SOLVE(eqn, x, -1, -0.5) User #110:[x = -0.784966] Simp(#109) #111:SOLVE(eqn, x, -0.5, 0.5) User #112:[x = -0.0162907] Simp(#111) #113:SOLVE(eqn, x, 0.5, 1) User #114:[x = 0.802556] Simp(#113) #115:Precision := Exact User #116:eqn := User #117:" x = {-1, 3} " User #118:SOLVE(|x - 1| = 2, x) User #119:[x = -1, x = 3] Simp(#118) #120:" x = {-1, -7} " User #121:SOLVE(|2*x + 5| = |x - 2|, x) User #122:[x = -1, x = -7] Simp(#121) #123:" x = +-3/2 " User #124:SOLVE(1 - |x| = MAX(-x - 2, x - 2), x) User / 3 3 \ #125:|x = ---, x = - ---| Simp(#124) \ 2 2 / #126:" x = {-1, 3} " User / 3 \ 2 | x | #127:eqn := MAX(2 - x , x) = MAX|-x, ----| User \ 9 / #128:SOLVE(eqn, x) User Simp(#128) / | 2 | 2 3 2 \ #129:\9*|x + x - 2| - (x + 9)*|x| - x - 9*x + 18*x = -18/ #130:Precision := Approximate User #131:SOLVE(eqn, x) User #132:[x = 3] Simp(#131) #133:Precision := Exact User #134:eqn := User User #135:" x = {+-3, -3 [1 + sqrt(3) sin t + cos t]} = {+-3, -1.554894} " User #136:" where t = (arctan[sqrt(5)/2] - pi)/3. The third answer is the root of " #137:" x^3 + 9 x^2 - 18 = 0 in the interval (-2, -1). " User 3 2 x #138:eqn := MAX(2 - x , x) = ---- User 9 #139:SOLVE(eqn, x) User / | 2 | 3 2 \ #140:\9*|x + x - 2| - 2*x - 9*x + 9*x = -18/ Simp(#139) #141:Precision := Approximate User #142:SOLVE(eqn, x) User #143:[x = 3] Simp(#142) #144:Precision := Exact User #145:eqn := User #146:" z = 2 + 3 i " User #147:z :epsilon Complex User #148:z Simp(#147) #149:eqn := (1 + #i)*z + (2 - #i)*CONJ(z) = - 3*#i User #150:SOLVE(eqn, z) User Simp(#150) / 3 3 | z #i*z #151:|SIGN(z) = #i*inf, SIGN(z) = - #i*inf, ------ + 2*z + ------- - | 2 2 \ |z| |z| \ 2 | #i*z + 3*#i*SIGN(z) = 0| | / #152:z := User #153:x :epsilon Real User #154:x Simp(#153) #155:y :epsilon Real User #156:y Simp(#155) User #157:eqn := (1 + #i)*(x + #i*y) + (2 - #i)*CONJ(x + #i*y) = - 3*#i #158:eqn User #159:3*x - 2*y - #i*y = - 3*#i Simp(#158) #160:SOLVE(eqn, [x, y]) User / 2*y #i*(y - 3) \ #161:|x = ----- + ------------| Simp(#160) \ 3 3 / #162:eqn := User #163:x := User #164:y := User #165:" => {f^(-1)(1), f^(-1)(-2)} assuming f is invertible "User #166:F(x) := User 2 #167:SOLVE(F(x) + F(x) - 2 = 0, x) User #168:[F(x) = 1, F(x) = -2] Simp(#167) #169:f := User #170:eqns := User #171:vars := User #172:" Solve a 3 x 3 system of linear equations " User #173:eqn1 := x + y + z = 6 User #174:eqn2 := 2*x + y + 2*z = 10 User #175:eqn3 := x + 3*y + z = 10 User User #176:" Note that the solution is parametric: x = 4 - z, y = 2 " #177:SOLVE([eqn1, eqn2, eqn3], [x, y, z]) User #178:[ x = @1 y = 2 z = 4 - @1 ] Simp(#177) User #179:" A linear system arising from the computation of a truncated power series " User #180:" solution to a differential equation. There are 189 equations to be solved " User #181:" for 49 unknowns. 42 of the equations are repeats of other equations; many " User #182:" others are trivial. Solving this system directly by Gaussian elimination " User #183:" is *not* a good idea. Solving the easy equations first is probably a better " User #184:" method. The solution is actually rather simple. [Stanly Steinberg] " User #185:" => k1 = ... = k22 = k24 = k25 = k27 = ... = k30 = k32 = k33 = k35 = ... " User #186:" = k38 = k40 = k41 = k44 = ... = k49 = 0, k23 = k31 = k39, " User #187:" k34 = b/a k26, k42 = c/a k26, {k23, k26, k43} are arbitrary " User / | b*k8 c*k8 b*k11 c*k11 #188:eqns := |- ------ + ------ = 0, - ------- + ------- = 0, - | a a a a \ b*k10 c*k10 b*k9 c*k9 ------- + ------- + k2 = 0, -k3 - ------ + ------ = 0, - a a a a b*k14 c*k14 b*k15 c*k15 b*k18 ------- + ------- = 0, - ------- + ------- = 0, - ------- + a a a a a c*k18 b*k17 c*k17 b*k16 c*k16 ------- - k2 = 0, - ------- + ------- = 0, - ------- + ------- a a a a a b*k13 c*k13 b*k21 c*k21 b*k5 + k4 = 0, - ------- + ------- - ------- + ------- + ------ - a a a a a c*k5 b*k44 c*k44 b*k45 c*k45 ------ = 0, ------- - ------- = 0, - ------- + ------- = 0, - a a a a a b*k20 c*k20 b*k44 c*k44 b*k46 ------- + ------- = 0, - ------- + ------- = 0, ------- - a a a a a 2 2 c*k46 b *k47 2*b*c*k47 c *k47 ------- = 0, -------- - ----------- + -------- = 0, k3 = 0, a 2 2 2 a a a b*k12 c*k12 a*k6 c*k6 b*k19 -k4 = 0, - ------- + ------- - ------ + ------ = 0, - ------- a a b b a c*k19 a*k7 b*k7 b*k45 c*k45 + ------- + ------ - ------ = 0, ------- - ------- = 0, - a c c a a 2 b*k46 c*k46 c*k48 c*k48 c *k48 ------- + ------- = 0, -k48 + ------- + ------- - -------- = a a a b a*b 2 b*k49 b*k49 b *k49 a*k1 c*k1 0, -k49 + ------- + ------- - -------- = 0, ------ - ------ = a c a*c b b a*k4 c*k4 a*k3 c*k3 0, ------ - ------ = 0, ------ - ------ + k9 = 0, -k10 + b b b b a*k2 c*k2 a*k7 c*k7 ------ - ------ = 0, ------ - ------ = 0, -k9 = 0, k11 = 0, b b b b b*k12 c*k12 a*k6 c*k6 a*k15 c*k15 ------- - ------- + ------ - ------ = 0, ------- - ------- = a a b b b b a*k18 c*k18 a*k17 c*k17 0, k10 + ------- - ------- = 0, -k11 + ------- - ------- = 0, b b b b a*k16 c*k16 a*k13 c*k13 a*k21 c*k21 ------- - ------- = 0, - ------- + ------- + ------- - ------- b b b b b b a*k5 c*k5 a*k44 c*k44 a*k45 + ------ - ------ = 0, - ------- + ------- = 0, ------- - b b b b b c*k45 a*k14 b*k14 a*k20 c*k20 ------- = 0, ------- - ------- + ------- - ------- = 0, b c c b b a*k44 c*k44 a*k46 c*k46 c*k47 ------- - ------- = 0, - ------- + ------- = 0, -k47 + ------- b b b b a 2 c*k47 c *k47 a*k19 c*k19 a*k45 + ------- - -------- = 0, ------- - ------- = 0, - ------- + b a*b b b b 2 c*k45 a*k46 c*k46 a *k48 2*a*c*k48 ------- = 0, ------- - ------- = 0, -------- - ----------- + b b b 2 2 b b 2 2 c *k48 a*k49 a*k49 a *k49 -------- = 0, -k49 + ------- + ------- - -------- = 0, k16 = 2 b c b*c b a*k1 b*k1 a*k4 b*k4 0, -k17 = 0, - ------ + ------ = 0, -k16 - ------ + ------ = c c c c a*k3 b*k3 a*k2 b*k2 b*k19 0, - ------ + ------ = 0, k18 - ------ + ------ = 0, ------- - c c c c a c*k19 a*k7 b*k7 a*k6 b*k6 a*k8 ------- - ------ + ------ = 0, - ------ + ------ = 0, - ------ a c c c c c b*k8 a*k11 b*k11 a*k10 + ------ = 0, - ------- + ------- + k17 = 0, - ------- + c c c c b*k10 a*k9 b*k9 a*k14 b*k14 ------- - k18 = 0, - ------ + ------ = 0, - ------- + ------- c c c c c a*k20 c*k20 a*k13 b*k13 a*k21 - ------- + ------- = 0, - ------- + ------- + ------- - b b c c c b*k21 a*k5 b*k5 a*k44 b*k44 ------- - ------ + ------ = 0, ------- - ------- = 0, - c c c c c a*k45 b*k45 a*k44 b*k44 a*k46 ------- + ------- = 0, - ------- + ------- = 0, ------- - c c c c c 2 b*k46 b*k47 b*k47 b *k47 ------- = 0, -k47 + ------- + ------- - -------- = 0, - c a c a*c a*k12 b*k12 a*k45 b*k45 a*k46 ------- + ------- = 0, ------- - ------- = 0, - ------- + c c c c c 2 2 b*k46 a*k48 a*k48 a *k48 a *k49 ------- = 0, -k48 + ------- + ------- - -------- = 0, -------- c b c b*c 2 c 2 2*a*b*k49 b *k49 - ----------- + -------- = 0, k8 = 0, k11 = 0, -k15 = 0, k10 - 2 2 c c k18 = 0, -k17 = 0, k9 = 0, -k16 = 0, -k29 = 0, k14 - k32 = 0, -k21 + k23 - k31 = 0, -k24 - k30 = 0, -k35 = 0, k44 = 0, -k45 = 0, k36 = 0, k13 - k23 + k39 = 0, -k20 + k38 = 0, k25 + k37 = b*k26 c*k26 0, ------- - ------- - k34 + k42 = 0, - 2*k44 = 0, k45 = 0, a a b*k47 c*k47 k46 = 0, ------- - ------- = 0, k41 = 0, k44 = 0, -k46 = 0, - a a b*k47 c*k47 ------- + ------- = 0, k12 + k24 = 0, -k19 - k25 = 0, - a a a*k27 c*k27 a*k48 ------- + ------- - k33 = 0, k45 = 0, -k46 = 0, - ------- + b b b c*k48 a*k28 b*k28 ------- = 0, ------- - ------- + k40 = 0, -k45 = 0, k46 = 0, b c c a*k48 c*k48 a*k49 b*k49 a*k49 ------- - ------- = 0, ------- - ------- = 0, - ------- + b b c c c b*k49 ------- = 0, -k1 = 0, -k4 = 0, -k3 = 0, k15 = 0, k18 - k2 = 0, c k17 = 0, k16 = 0, k22 = 0, k25 - k7 = 0, k24 + k30 = 0, k21 + k23 - k31 = 0, k28 = 0, -k44 = 0, k45 = 0, -k30 - k6 = 0, k20 b*k33 c*k33 + k32 = 0, k27 + ------- - ------- = 0, k44 = 0, -k46 = 0, - a a b*k47 c*k47 ------- + ------- = 0, -k36 = 0, k31 - k39 - k5 = 0, -k32 - a a a*k34 c*k34 k38 = 0, k19 - k37 = 0, k26 - ------- + ------- - k42 = 0, k44 b b a*k48 c*k48 a*k35 = 0, - 2*k45 = 0, k46 = 0, ------- - ------- = 0, ------- - b b c b*k35 b*k47 c*k47 ------- - k41 = 0, -k44 = 0, k46 = 0, ------- - ------- = 0, - c a a a*k49 b*k49 a*k48 ------- + ------- = 0, -k40 = 0, k45 = 0, -k46 = 0, - ------- c c b c*k48 a*k49 b*k49 + ------- = 0, ------- - ------- = 0, k1 = 0, k4 = 0, k3 = 0, b c c -k8 = 0, -k11 = 0, -k10 + k2 = 0, -k9 = 0, k37 + k7 = 0, -k14 - k38 = 0, -k22 = 0, -k25 - k37 = 0, -k24 + k6 = 0, -k13 - k23 b*k40 c*k40 + k39 = 0, -k28 + ------- - ------- = 0, k44 = 0, -k45 = 0, a a b*k47 c*k47 -k27 = 0, -k44 = 0, k46 = 0, ------- - ------- = 0, k29 = 0, a a k32 + k38 = 0, k31 - k39 + k5 = 0, -k12 + k30 = 0, k35 - a*k41 c*k41 a*k42 ------- + ------- = 0, -k44 = 0, k45 = 0, -k26 + k34 + ------- b b c b*k42 b*k47 - ------- = 0, k44 = 0, k45 = 0, - 2*k46 = 0, - ------- + c a c*k47 a*k48 c*k48 a*k49 b*k49 ------- = 0, - ------- + ------- = 0, ------- - ------- = 0, a b b c c a*k48 c*k48 a*k49 k33 = 0, -k45 = 0, k46 = 0, ------- - ------- = 0, - ------- + b b c \ b*k49 | ------- = 0| c | / User #189:vars := [k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13, k14, k15, k16, k17, k18, k19, k20, k21, k22, k23, k24, k25, k26, k27, k28, k29, k30, k31, k32, k33, k34, k35, k36, k37, k38, k39, k40, k41, k42, k43, k44, k45, k46, k47, k48, k49] #190:SOLVE(eqns, vars) User Simp(#190) / #191:| k1 = 0 k2 = 0 k3 = 0 k4 = 0 k5 = 0 k6 = 0 k7 = 0 k8 = 0 \ k9 = 0 k10 = 0 k11 = 0 k12 = 0 k13 = 0 k14 = 0 k15 = 0 k16 = 0 k17 = 0 k18 = 0 k19 = 0 k20 = 0 k21 = 0 k22 = 0 k23 = @2 k24 = 0 k25 = 0 k26 = @3 k27 = 0 k28 = 0 k29 = @3*b 0 k30 = 0 k31 = @2 k32 = 0 k33 = 0 k34 = ------ k35 = 0 a k36 = 0 k37 = 0 k38 = 0 k39 = @2 k40 = 0 k41 = 0 k42 = @3*c ------ k43 = @4 k44 = 0 k45 = 0 k46 = 0 k47 = 0 k48 = 0 a \ k49 = 0 | / #192:" Solve a 3 x 3 system of nonlinear equations " User 2 #193:eqn1 := x *y + 3*y*z - 4 = 0 User 2 2 #194:eqn2 := - 3*x *z + 2*y + 1 = 0 User 2 2 #195:eqn3 := 2*y*z - z - 1 = 0 User #196:" Solving this by hand would be a nightmare " User #197:SOLVE([eqn1, eqn2, eqn3], [x, y, z]) User #198:[] Simp(#197) #199:Precision := Approximate User #200:SOLVE([eqn1, eqn2, eqn3], [x, y, z]) User #201:[] Simp(#200) #202:Precision := Exact User #203:eqn1 := User #204:eqn2 := User #205:eqn3 := User #206:" ---------- Quit ---------- " User