Lesson Solving system of linear equation in 19 unknowns
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<H2>Solving system of linear equation in 19 unknowns</H2> You can consider this problem as a Math joke, or as a Math entertainment, or seriously. In any case, my goal is to teach you. <H3>Problem 1</H3>If {{{a[1]}}}, {{{a[2]}}}, . . ., {{{a[19]}}} satisfy <pre> {{{a[1] + a[2] + a[3] + a[4]}}} = 1, {{{a[2] + a[3] + a[4] + a[5]}}} = 2, {{{a[3] + a[4] + a[5] + a[6]}}} = 3, . . . . . . . . . {{{a[16] + a[17] + a[18] + a[19]}}} = 16, {{{a[17] + a[18] + a[19] + a[1]}}} = 17, {{{a[18] + a[19] + a[1] + a[2]}}} = 18, {{{a[19] + a[1] + a[2] + a[3]}}} = 19, </pre>find the value of {{{a[19]}}}. <B>Solution</B> <pre> Let's write the system of equations in full {{{a[1] + a[2] + a[3] + a[4]}}} = 1 (1) {{{a[2] + a[3] + a[4] + a[5]}}} = 2 (2) {{{a[3] + a[4] + a[5] + a[6]}}} = 3 (3) . . . . . . . . . {{{a[16] + a[17] + a[18] + a[19]}}} = 16 (16) {{{a[17] + a[18] + a[19] + a[1]}}} = 17 (17) {{{a[18] + a[19] + a[1] + a[2]}}} = 18 (18) {{{a[19] + a[1] + a[2] + a[3]}}} = 19 (19) Subtract equation (1) from equation (2): {{{(a[2] + a[3] + a[4] + a[5])}}} - {{{(a[1] + a[2] + a[3] + a[4])}}} = 2 - 1 {{{a[5] - a[1]}}} = 1 {{{a[5]}}} = {{{a[1]}}} + 1 Subtract equation (2) from equation (3): {{{(a[3] + a[4] + a[5] + a[6])}}} - {{{(a[2] + a[3] + a[4] + a[5])}}} = 3 - 2 {{{a[6]}}} - {{{a[2]}}} = 1 {{{a[6]}}} = {{{a[2]}}} + 1 In general, we have {{{a[n+4]}}} = {{{a[n] + 1}}}. This means: {{{a[5]}}} = {{{a[1]}}} + 1 {{{a[9]}}} = {{{a[5] + 1}}} = {{{a[1] + 2}}} {{{a[13]}}} = {{{a[9] + 1}}} = {{{a[1] + 3}}} {{{a[17]}}} = {{{a[13] + 1}}} = {{{a[1] + 4}}} Similarly, {{{a[6]}}} = {{{a[2] + 1}}} {{{a[10]}}} = {{{a[6] + 1}}} = {{{a[2] + 2}}} {{{a[14]}}} = {{{a[10] + 1}} = {{{a[2] + 3}}} {{{a[18]}}} = {{{a[14] + 1}}} = {{{a[2] + 4}}} also, {{{a[7]}}} = {{{a[3] + 1}}} {{{a[11]}}} = {{{a[7] + 1}}} = {{{a[3] + 2}}} {{{a[15]}}} = {{{a[11] + 1}}} = {{{a[3] + 3}}} {{{a[19]}}} = {{{a[15] + 1}}} = {{{a[3] + 4}}} and {{{a[8]}}} = {{{a[4] + 1}}} {{{a[12]}}} = {{{a[8] + 1}}} = {{{a[4] + 2}}} {{{a[16]}}} = {{{a[12] + 1}}} = {{{a[4] + 3}}} Now let's use equations (1), (17), (18), and (19). From equation (1), {{{a[1] + a[2] + a[3] + a[4]}}} = 1 (20) From equation (17), {{{a[17] + a[18] + a[19] + a[1]}}} = 17 (21) From equation (18), {{{a[18] + a[19] + a[1] + a[2]}}} = 18 (22) From equation (19), {{{a[19] + a[1] + a[2] + a[3]}}} = 19 (23) Substituting the expressions we find: from (20): {{{(a[1] + 4) + (a[2] + 4) + (a[3] + 4) + a[1]}}} = 17 {{{2a[1] + a[2] + a[3] + 12}}} = 17 {{{2a[1] + a[2] + a[3]}}} = 5 (17') from (21): {{{(a[2] + 4) + (a[3] + 4) + a[1] + a[2]}}} = 18 {{{a[1] + 2a[2] + a[3] + 8}}} = 18 {{{a[1] + 2a[2] + a[3]}}} = 10 (18') from (22): {{{(a[3] + 4) + a[1] + a[2] + a[3]}}} = 19 {{{a[1] + a[2] + 2a[3] + 4}}} = 19 {{{a[1] + a[2] + 2a[3]}}} = 15 (19') Add equations (17'), (18') and (19'). You will get {{{4a[1] + 4a[2] + 4a[3]}}} = 5 + 10 + 15 {{{4a[1] + 4a[2] + 4a[3]}}} = 30. Divide both sides by 4 {{{a[1] + a[2] + a[3]}}} = 7.5. (24) Now we are on the finish line. From equation (17') subtract equation (24). You will get {{{a[1]}}} = 5 - 7.5 = -2.5. From equation (18') subtract equation (24). You will get {{{a[2]}}} = 10 - 7.5 = 2.5. From equation (19') subtract equation (24). You will get {{{a[3]}}} = 15 - 7.5 = 7.5. From equation (23), {{{a[19]}}} = 19 - {{{a[1]}}} - {{{a[2]}}} - {{{a[3]}}} = 19 - 0 - (-2.5) - 2.5 - 7.5 = 11.5 For completeness, let's determine a-1 from the very first equation in this post {{{a[1]}}} = {{{1 - (a[2] + a[3] + a[4])}}} = 1 - ((-2.5) + 2.5 + 7.5) = 1 - 7.5 = -6.5. From this point, all 19 terms {{{a[1]}}}, {{{a[2]}}}, {{{a[3]}}}, {{{a[4]}}}, {{{a[5]}}}, . . . , {{{a[19]}}} can be determined. See this table below a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8 a_9 a_10 a_11 a_12 a_13 a_14 a_15 a_16 a_17 a_18 a_19 -2.5 2.5 7.5 -6.5 -1.5 3.5 8.5 -5.5 -0.5 4.5 9.5 -4.5 0.5 5.5 10.5 5.5 1.5 6.5 11.5 You may check that all given 19 original equations are satisfied. Final Answer: {{{a[19]}}} = 11.5. </pre> My other lessons in this site on determinants of 3x3-matrices and the Cramer's rule for solving systems of linear equations in three unknowns are - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Determinant-of-a-3x3-matrix.lesson>Determinant of a 3x3 matrix</A> - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Co-factoring-a-3x3-determinant.lesson>Co-factoring the determinant of a 3x3 matrix</A> - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/HOW-TO-solve-system-of-linear-eqns-in-three-unknowns-using-det.lesson>HOW TO solve system of linear equations in three unknowns using determinant (Cramer's rule)</A> - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-linear-equations-in-three-unknowns-using-determinant.lesson>Solving systems of linear equations in three unknowns using determinant (Cramer's rule)</A> - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-word-problems-by-reducing-them-to-systems-of-linear-equations-in-three-unknowns.lesson>Solving word problems by reducing to systems of linear equations in three unknowns</A> - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/The-trick-to-solve-some-word-problems-with-three-and-more-unknowns.lesson>The tricks to solve some word problems with three and more unknowns using mental math</A> - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-non-linear-equations-in-three-unknowns-using-Cramer%27s-rule.lesson>Solving systems of non-linear equations in three unknowns using Cramer's rule</A> - <A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Sometime-two-eqns-are-enough-to-find-three-unknowns-by-an-UNIQUE-way.lesson>Sometime two equations are enough to find three unknowns by an UNIQUE way</A> - <A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Two-very-different-approaches-to-one-word-problem.lesson>Two very different approaches to one word problem</A> - <A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-word-problems-in-three-unknowns-by-the-backward-method.lesson>Solving word problems in three unknowns by the backward method</A> - <A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-a-system-of-linear-equations-in-17-unknowns.lesson>Solving system of linear equation in 17 unknowns</A> - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/OVERVIEW-of-LESSONS-on-dets-of-3x3-matrices-and-Cramer%27s-rule-for-systems-in-3-unknowns.lesson>OVERVIEW of LESSONS on determinants of 3x3-matrices and Cramer's rule for systems in 3 unknowns</A> under the current topic <B>Matrices, determinant, Cramer rule</B> of the section <B>Algebra-II</B>. My other lessons in this site on solving systems of linear equations in three unknowns are - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-linear-equations-in-3-unknowns-by-the-Substitution-method.lesson>Solving systems of linear equations in 3 unknowns by the Substitution method</A>, - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/BRIEFLY-on-solving-systems-of-linear-eqns-in-3-unknowns-by-the-Subst-method.lesson>BRIEFLY on solving systems of linear equations in 3 unknowns by the Substitution method</A>, - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-linear-equations-in-3-unknowns-by-the-Elimination-method.lesson>Solving systems of linear equations in 3 unknowns by the Elimination method</A> and - <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/BRIEFLY-on-solving-systems-of-linear-eqns-in-3-unknowns-by-the-Eliminat-method.lesson>BRIEFLY on solving systems of linear equations in 3 unknowns by the Elimination method</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
