Notes: `8x - 4y = 7, 5x + 2y = 1` Use matricies to solve the system of equations (if possible). Use Gauss-Jordan elimination.

Sunday, January 22, 2017

Given system of equations are

$\displaystyle{8}{x}-{4}{y}={7},{5}{x}+{2}{y}={1}$

so the matrices A and B are given as follows

A = $\displaystyle{\left[\matrix{{8}&-{4}\\{5}&{2}}\right]}$

B = $\displaystyle{\left[\matrix{{7}\\{1}}\right]}$

so the augmented matrix is [AB] = $\displaystyle{\left[\matrix{{8}&-{4}&{7}\\{5}&{2}&{1}}\right]}$

on solving this we get the values of x,y .

step 1 . Make the pivot in the 1st column by dividing the 1st row by 8

$\displaystyle{\left[{\left[{1},-\frac{{1}}{{2}},\frac{{7}}{{8}}\right]},\ldots\right.}$

Given system of equations are

$\displaystyle{8}{x}-{4}{y}={7},{5}{x}+{2}{y}={1}$

so the matrices A and B are given as follows

A = $\displaystyle{\left[\matrix{{8}&-{4}\\{5}&{2}}\right]}$

B = $\displaystyle{\left[\matrix{{7}\\{1}}\right]}$

so the augmented matrix is [AB] = $\displaystyle{\left[\matrix{{8}&-{4}&{7}\\{5}&{2}&{1}}\right]}$

on solving this we get the values of x,y .

step 1 . Make the pivot in the 1st column by dividing the 1st row by 8

$\displaystyle{\left[\matrix{{1}&-\frac{{1}}{{2}}&\frac{{7}}{{8}}\\{5}&{2}&{1}}\right]}$

step 2 . muptiply the 1st row by 5

$\displaystyle{\left[\matrix{{5}&-\frac{{5}}{{2}}&\frac{{35}}{{8}}\\{5}&{2}&{1}}\right]}$

step 3 . subtract the 1st row from the 2nd row

$\displaystyle{\left[\matrix{{1}&-\frac{{1}}{{2}}&\frac{{7}}{{8}}\\{0}&\frac{{9}}{{2}}&-\frac{{27}}{{8}}}\right]}$

step 4 divide the second row by 9/2

$\displaystyle{\left[\matrix{{1}&-\frac{{1}}{{2}}&\frac{{7}}{{8}}\\{0}&{1}&-\frac{{3}}{{4}}}\right]}$

step 5 multiply the 2nd row by -1/2 and subtract the 2nd row from the 1st row

$\displaystyle{\left[\matrix{{1}&{0}&\frac{{1}}{{2}}\\{0}&{1}&-\frac{{3}}{{4}}}\right]}$

so, the values of x, y are x= 1/2 and y = -3/4

Notes