Lesson Solving system of linear equation in 17 unknowns
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<H2>Solving system of linear equation in 17 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[17]}}}) satisfy <pre> {{{a[1] + a[2] + a[3]}}} = 1, {{{a[2] + a[3] + a[4]}}} = 2, {{{a[3] + a[4] + a[5]}}} = 3, . . . . . . . . . {{{a[15] + a[16] + a[17]}}} = 15, {{{a[16] + a[17] + a[1]}}} = 16, {{{a[17] + a[1] + a[2]}}} = 17, </pre>find the value of {{{a[17]}}}. <B>Solution</B> <pre> Let's write the system of equations in full {{{a[1] + a[2] + a[3]}}} = 1, (1) {{{a[2] + a[3] + a[4]}}} = 2, (2) {{{a[3] + a[4] + a[5]}}} = 3, (3) {{{a[4] + a[5] + a[6]}}} = 4, (4) {{{a[5] + a[6] + a[7]}}} = 5, (5) {{{a[6] + a[7] + a[8]}}} = 6, (6) {{{a[7] + a[8] + a[9]}}} = 7, (7) {{{a[8] + a[9] + a[10]}}} = 8, (8) {{{a[9] + a[10] + a[11]}}} = 9, (9) {{{a[10] + a[11] + a[12]}}} = 10, (10) {{{a[11] + a[12] + a[13]}}} = 11, (11) {{{a[12] + a[13] + a[14]}}} = 12, (12) {{{a[13] + a[14] + a[15]}}} = 13, (13) {{{a[14] + a[15] + a[16]}}} = 14, (14) {{{a[15] + a[16] + a[17]}}} = 15, (15) {{{a[16] + a[17] + a[1]}}} = 16, (16) {{{a[17] + a[1] + a[2]}}} = 17. (17) Add all 17 equations (1) - (17) (both sides). You will get {{{3*(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])}}} = 1 + 2 + 3 + . . . + 15 + 16 + 17 = {{{((1 + 17)/2)*17}}} = 9*17 = 153. Now divide both sides of the lest equation by 3. You will get {{{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]}}} = 51. (18) Next subtract equation (1), (4), (7), (10), (13) from equation (18). You will get {{{a[16]+a[17]}}} = 51 - 1 - 4 - 7 - 10 - 13 = 16. (19) Now compare equations (19) and (16). You instantly will get {{{a[1]}}} = 0. (20) Having known {{{a[1]}}} = 0, we can rewrite the equation (20) in the form {{{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]}}} = 51. (21) </pre> <U>Now we are on the finish line, finally !!!</U> <pre> Next subtract equation (2), (5), (8), (11) from equation (21). You will get {{{a[16]+a[17]}}} = 51 - 2 - 5 - 8 - 11 = 25. (22) As the last step, subtract equation (14) from equation (22). You will get {{{a[17]}}} = 25 - 14 = 11. (30) It is your answer: {{{a[17]}}} = 11. For completeness, moving from equation (1) to equation (17), we can determine all 17 unknowns: 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 0 6 -5 1 7 -4 2 8 -3 3 9 -2 4 10 -1 5 11 You may check that all given 17 original equations are satisfied. </pre> At this point, the problem is fully solved. <U>Answer</U>. {{{a[17]}}} = 11. 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-19-unknowns.lesson>Solving system of linear equation in 19 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.
