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Given the $5$ equations.

$a_1=a_2+a_5 \tag{1}$

$a_4=a_3+a_5 \tag{2}$

$b=a_1+a_2 \tag{3}$

$b=a_3+\frac{a_4}{2} \tag{4}$

$\frac{a_5}{2}+a_1=a_3 \tag{5}$

I want to deduce the following.

$a_1=\frac{9}{16}*b\tag{6}$

$a_2=\frac{7}{16}*b\tag{7}$

$a_3=\frac{5}{8}*b\tag{8}$

$a_4=\frac{3}{4}*b\tag{9}$

$a_5=\frac{1}{8}*b\tag{10}$

Can anyone tell some hint(s)? so that I can deduce it in my own.

  • Treat $b$ as if it were a constant. Then you have $5$ linear equations in the $5$ unknowns $a_1,\dots,a_5$ – saulspatz Mar 03 '21 at 04:41
  • There are standard methods for solving systems of linear equations (which is what you have here). Those methods involve setting up a matrix to represent the system, and using row operations to reduce the matrix to echelon form. You can find details in any introductory Linear Algebra textbook. – Gerry Myerson Mar 03 '21 at 05:50
  • Or, you can just use each equation to eliminate one variable from all the following equations. Use (1) to eliminate $a_1$, then (2) to eliminate $a_4$, and so on. – Gerry Myerson Mar 03 '21 at 05:52
  • I'll try a matrix method. –  Mar 03 '21 at 06:02

1 Answers1

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Hint 1: As the comments have stated, one approach is to write the system as a matrix and then find the RREF. The matrix is given by $$A = \left( \begin{array}{cccccc} 1 & -1 & 0 & 0 & -1 & 0 \\ 0 & 0 & 1 & -1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & b \\ 0 & 0 & 1 & \frac{1}{2} & 0 & b \\ 1 & 0 & -1 & 0 & \frac{1}{2} & 0 \\ \end{array} \right)$$

You already have the result in your question, so know what the RREF should produce.

Hint 2: A second approach is to eliminate equations.

Moo
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