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Number of Solutions of Equations

Number of Solutions of Equations
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2023-12-27 05:36
Geometric Construction
This example demonstrates how to determine the number of solutions for linear equations (0, 1, or infinitely many). The key mathematical concept is the nature of solutions to linear equations. Construction involves using GeoGebra commands such as RandomUniform, Join, Flatten, etc. It is suitable for students in grades 6-8 for classroom practice and online learning resources.

GeoGebra Commands & Parameters

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.8387831301798795, 0.27662096401157443, 0.20104052179015452, 0.8540731639812609, 0.39625459387894063, 0.949690959227446, 0.6183012573506799, 0.6245131995893908, 0.2421232775644254, 0.8427134743495948, 0.9469186345703609, 0.15805114028411865, 0.10030726159112921, 0.2884616437027424, 0.5971024112408369, 0.48941622268703444, 0.0681484074382333, 0.44528727995424244, 0.13783309240856517, 0.815348799461414, 0.27325565798726276, 0.512995139726211, 0.10463354907274731, 0.033890593366123586, 0.954965987219833, 0.9240248301024554, 0.8429281880943177, 0.9748492480863082, 0.9857395749382619, 0.029952889278838876, 0.2987082540289937, 0.09109299426935417, 0.08428123594684471, 0.5181602612669733, 0.5407902602666259, 0.6550081913474816, 0.03834178186207434, 0.12553297259572496, 0.30620971162969673, 0.6213186190231497, 0.8710026813650416, 0.4352722515218295, 0.6822367013212384, 0.01651734274440818, 0.03978459494342361, 0.9392868939431277, 0.03512988001851574, 0.49640264177359544, 0.7131449307202373, 0.622224666902354, 0.46706421499516565, 0.8566372880743552, 0.11384152833617708, 0.15397115680616402, 0.3586735663336011, 0.06686520743086367, 0.4432396066407631, 0.19540333170724516, 0.9653425562290541, 0.8567229964752476, 0.6549253641360065, 0.5773315640106936, 0.6907631125307498, 0.6220733732646857, 0.16996553595568561, 0.41489775999138623, 0.42133928149288913, 0.06196561754106772, 0.29981299320502824, 0.36921917038498375, 0.15977985503068876, 0.05119352469435983, 0.8962088256653133, 0.4147757598007, 0.43566185281676595, 0.7098737203935592, 0.17046851853366968, 0.3049359636301114, 0.9148782607117568, 0.10020758527473039, 0.8572639360373681, 0.8990997695259443, 0.39870269009687487, 0.5815178866416715, 0.00042526088836703613, 0.3432088248248809, 0.4667084389271997, 0.27776643308050547, 0.5203868939945288, 0.5708332436108525, 0.3659734190294696, 0.09706145168484767, 0.704687590472579, 0.5846903531833761, 0.801883426265357, 0.2401541901658617, 0.7344110560396185, 0.008285320025167006, 0.14341388711474098, 0.06460147594131391, 0.48061360052953767, 0.17438464179821367, 0.8780267298257886, 0.06313653913908789, 0.5456369695682586, 0.2312425219263812, 0.4182622762201854, 0.898842954363642, 0.48473970151179235, 0.8188969504152798, 0.7337363265755998, 0.925966740905291, 0.45370673443918125, 0.2764357923054101, 0.13857202429081414, 0.5136298073692828, 0.7380076575489549, 0.05690517757114925, 0.15530090697808016, 0.07032250024083904, 0.29867428825875714, 0.32949110509084745, 0.43196147901167586, 0.21985625901040085, 0.8193555937792656, 0.25875187898494767, 0.8295045088253881, 0.8744475671587357, 0.027981995121703674, 0.3780126374819821, 0.8516822689248092, 0.48778176933089323, 0.3551333413788179, 0.17855537625436366, 0.3693552188262777, 0.7202710068224318, 0.9495551161620992, 0.43794613477102523, 0.7155924844558172, 0.3602730603502498, 0.499303361187143, 0.47973788400327044, 0.5070740657523766, 0.7587779153261446, 0.37472457853704605, 0.7328716810830399, 0.7311202191637639, 0.690996970579897, 0.9168882102250521, 0.4523697969927208, 0.5161799818480255, 0.7464049050311578, 0.5221891860086929, 0.45337394248243573, 0.3468038808408078, 0.7390868587673805, 0.8906252978064249, 0.018227954701653327, 0.6320067276488751, 0.7827338196049536, 0.33172824095256853, 0.9011381886173233, 0.3784948449034909, 0.6693069777489843, 0.5482900504374137, 0.5769667224700079, 0.8874878560682673, 0.3472570983416235, 0.46387546279532077, 0.2599209190862927, 0.9058085720351837, 0.9742372514392253, 0.4356483723501729, 0.4743000828904015, 0.8285454264947275, 0.1304881687735866, 0.5292651358294389, 0.1530431130319934, 0.5219100354363727, 0.972502588170903, 0.8774991741890665, 0.9205933331305955, 0.8709186849050158, 0.5221905010592308, 0.508861452544875, 0.8345549355221575, 0.2972194690011663, 0.7645700264148942, 0.8367681886198317, 0.8157627515226261, 0.3677683100949425, 0.7669985187546359, 0.7537819042883201, 0.7187200446157119, 0.22512434816138263, 0.38513677656252915, 0.18911386750787237, 0.763881688955405, 0.5748925821778162, 0.0752039035650212}
  2. Join({{1 / 2, 1 / 5, 2 / 5}, (-1 * {1 / 2, 1 / 5, 2 / 5})}) → allowedFractions
  3. Flatten({Sequence[-10, 10] \ {0}, allowedFractions}) → bNums
  4. Flatten({Sequence[-10, 10] \ {0, 1}, allowedFractions}) → aNums
  5. Sequence(-5, 5) → sols
  6. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  7. Element(sols, floor((Length[sols] * currentRandomSeeds(6))) + 1) → sol
  8. Element(sols, floor((Length[sols] * currentRandomSeeds(5))) + 1) → int
  9. Element(aNums, 1 + floor((currentRandomSeeds(3) * Length[aNums]))) → a
  10. Flatten({Sequence[-10, 10] \ {0, 1, a}, allowedFractions}) → cNums
  11. Element(cNums, floor((currentRandomSeeds(6) * Length[cNums])) + 1) → c1
  12. If(numOfSols ≠ 1, Element[bNums, floor((currentRandomSeeds(4) * Length[bNums])) + 1], int - (a * sol)) → b
  13. If(caseNum_{1} ≟ 3, left3_{1}, left12_{1}) → A1
  14. Flatten({Sequence[-10, 10] \ {0, b}, allowedFractions}) → dNums
  15. Element(dNums, floor((currentRandomSeeds(8) * Length[dNums])) + 1) → d1
  16. If(numOfSols ≟ 1, c1, a) → c
  17. If(solutionsCurrent_{1} ≟ 3, b_{1}, solutionsCurrent_{1} ≟ 1, int_{1} - (c_{1} * sol_{1}), d1_{1}) → d_{1}
  18. If(caseNum_{1} ≟ 1, right1_{1}, right23_{1}) → B1
  19. TableText(Take[solLeft, If[case ≥ 2, 1, 2], If[numOfSols ≟ 1, 5, 4]], Sequence["=", kk, 1, 3 + If[case ≠ 1, 1, 0] + If[numOfSols ≟ 1, 1, 0]], Take[solRight, If[case ≥ 2, 1, 2], If[numOfSols ≟ 1, 5, 4]], "vrcl") → text1
  20. Textfield(inputB) → InputBoxB
  21. Polygon(Zip[PixelsToPoint[Corner[1], Corner[3], Corner[5], A], A, {(pixelsLeft, pixelsTop), (pixelsLeft, y(Corner[5]) - pixelsBottom), Corner[5] - (pixelsRight, pixelsBottom), (x(Corner[5]) - pixelsRight, pixelsTop)}]) → borderBox
  22. Corner(5) → layoutAppSize
  23. onboardingTextOffset(Corner[onboardingText, 1], Corner[onboardingText, 3], onboardingCase, Corner[1], Corner[3], Corner[5]) → OnboardingTextOffset
  24. Textfield(inputA) → InputBoxA
  25. Textfield(inputC) → InputBoxC
  26. If(numOfSols ≟ 3, b, numOfSols ≟ 1, int - (c * sol), d1) → d
  27. Numerator(a_{1} - c_{1}) → numer_{1}
  28. Denominator(a_{1} - c_{1}) → denom_{1}
  29. Zip((("$\textsf{" * tt) * " }$"), tt, {"Simple: \(ax+b = cx+d\)", "More complex: \(ax+b = c(x+d)\)", "Most complex: \(a(x+b) = c(x+d)\)"}) → options_{1}
  30. If(caseNum_{1} ≟ 3, left3_{1}, left12_{1}) → left_{1}
  31. If(caseNum_{1} ≟ 1, right1_{1}, right23_{1}) → right_{1}
  32. TableText(Take[solLeft_{1}, If[caseNum_{1} ≥ 2, 1, 2], If[solutionsCurrent_{1} ≟ 1, 5, 4]], Sequence["=", CLIPBOARDmagicSTRINGkk, 1, 3 + If[caseNum_{1} ≠ 1, 1, 0] + If[solutionsCurrent_{1} ≟ 1, 1, 0]], Take[solRight_{1}, If[caseNum_{1} ≥ 2, 1, 2], If[solutionsCurrent_{1} ≟ 1, 5, 4]], "vrcl") → solSimple2_{1}
  33. Translate(hGL, Vector[(Vector[Corner[1], Corner[4]] / 500 * 16)]) → hGL'

AI Drawing Prompt Reference

Create an interactive educational diagram for practicing the number of solutions of linear equations. The center shows an equation in the form ax+b=cx+d or with parentheses, with text boxes for coefficients and constants on both sides. Below the equation, input boxes allow the user to enter the solution, and the number of solutions (0, 1, or infinitely many) is displayed. The background is white with a border, and a title and option menu at the top let the user choose simple, medium, or complex problem types. All coefficients and constants are randomly selected from preset sets of integers and fractions to ensure variety.
The prompt above was compiled from the original drawing commands and is for reference only.