MHB Mason's question via Facebook about solving a system of equations

AI Thread Summary
The discussion revolves around solving a system of equations involving three variables: x, y, and z. The equations are manipulated using techniques such as finding the least common multiple (LCM) of coefficients, multiplying equations to align them, and applying row operations to simplify the system. The final solution obtained through these methods is x = 5, y = 8, and z = -6. The conversation also suggests that representing the solution in matrix form and using Gaussian elimination could be beneficial for further analysis. The solution process highlights the effectiveness of systematic approaches in solving linear equations.
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Solve the following system for $\displaystyle \begin{align*} x, y, z \end{align*}$:

$\displaystyle \begin{align*} 5\,x - 2\,y + z &= 3 \\ 3\,x + y + 3\,z &= 5 \\ 6\,x + y - 4\,z &= 62 \end{align*}$

The LCM of the $\displaystyle \begin{align*} x \end{align*}$ coefficients is 30, so multiplying the first equation by 6, the second by 10 and the third by 5 gives

$\displaystyle \begin{align*} 30\,x - 12\,y + 6\,z &= 18 \\ 30\,x + 10\,y + 30\,z &= 50 \\ 30\,x + 5\,y - 20\,z &= 310 \end{align*}$

Applying R2 - R1 to R2 and R3 - R1 to R3 we have

$\displaystyle \begin{align*} 30\,x + 12\,y - 6\,z &= 18 \\ 22\,y + 24\,z &= 32 \\ 17\,y - 26\,z &= 292 \end{align*}$

Dividing the second equation by 2 gives

$\displaystyle \begin{align*} 30\,x + 12\,y - 6\,z &= 18 \\ 11\,y + 12\,z &= 16 \\ 17\,y - 26 \,z &= 292 \end{align*}$

The LCM of the y coefficients in rows 2 and 3 is 187, so multiplying the second equation by 17 and the third equation by 11 we have

$\displaystyle \begin{align*} 30\,x + 12\,y - 6\,z &= 18 \\ 187\,y + 204\,z &= 272 \\ 187\,y - 286\,z &= 3\,212 \end{align*}$

Applying R3 - R2 to R2 we have

$\displaystyle \begin{align*} 30\,x + 12\,y - 6\,z &= 18 \\ 187\,y + 204\,z &= 272 \\ - 490\,z &= 2\,940 \end{align*}$

Since $\displaystyle \begin{align*} -490\,z = 2\,940 \implies z = -6 \end{align*}$, then

$\displaystyle \begin{align*} 187\,y + 204 \, \left( -6 \right) &= 272 \\ 187\,y - 1\,224 &= 272 \\ 187\,y &= 1\,496 \\ y &= 8 \end{align*}$

and

$\displaystyle \begin{align*} 5\,x - 2\,\left( 8 \right) + \left( -6 \right) &= 3 \\ 5\,x - 22 &= 3 \\ 5\,x &= 25 \\ x &= 5 \end{align*}$

So the solution is $\displaystyle \begin{align*} \left( x , y , z \right) = \left( 5, 8, -6 \right) \end{align*}$.
 
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The LCM of the $\displaystyle \begin{align*} x \end{align*}$ coefficients is 30, so multiplying the first equation by 6, the second by 10 and the third by 5 gives

$\displaystyle \begin{align*} 30\,x - 12\,y + 6\,z &= 18 \\ 30\,x + 10\,y + 30\,z &= 50 \\ 30\,x + 5\,y - 20\,z &= 310 \end{align*}$

Applying R2 - R1 to R2 and R3 - R1 to R3 we have

$\displaystyle \begin{align*} 30\,x + 12\,y - 6\,z &= 18 \\ 22\,y + 24\,z &= 32 \\ 17\,y - 26\,z &= 292 \end{align*}$

Dividing the second equation by 2 gives

$\displaystyle \begin{align*} 30\,x + 12\,y - 6\,z &= 18 \\ 11\,y + 12\,z &= 16 \\ 17\,y - 26 \,z &= 292 \end{align*}$

The LCM of the y coefficients in rows 2 and 3 is 187, so multiplying the second equation by 17 and the third equation by 11 we have

$\displaystyle \begin{align*} 30\,x + 12\,y - 6\,z &= 18 \\ 187\,y + 204\,z &= 272 \\ 187\,y - 286\,z &= 3\,212 \end{align*}$

Applying R3 - R2 to R2 we have

$\displaystyle \begin{align*} 30\,x + 12\,y - 6\,z &= 18 \\ 187\,y + 204\,z &= 272 \\ - 490\,z &= 2\,940 \end{align*}$

Since $\displaystyle \begin{align*} -490\,z = 2\,940 \implies z = -6 \end{align*}$, then

$\displaystyle \begin{align*} 187\,y + 204 \, \left( -6 \right) &= 272 \\ 187\,y - 1\,224 &= 272 \\ 187\,y &= 1\,496 \\ y &= 8 \end{align*}$

and

$\displaystyle \begin{align*} 5\,x - 2\,\left( 8 \right) + \left( -6 \right) &= 3 \\ 5\,x - 22 &= 3 \\ 5\,x &= 25 \\ x &= 5 \end{align*}$

So the solution is $\displaystyle \begin{align*} \left( x , y , z \right) = \left( 5, 8, -6 \right) \end{align*}$.
Correct. The next step would be to write it in form of matrices. 'Gaussian elimination' would be a suitable search key.
 
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