Linear System Solution Method and Validity

[yango_17]

Homework Statement

Find all solutions of the linear system
x + 2y + 3z = a
x + 3y + 8z = b
x + 2y + 2z = c
where a,b, and c are arbitrary constants.

Homework Equations

The Attempt at a Solution

Using elimination, I managed to set the coefficients on the diagonal equal to 1, which then allowed me to solve for z, which was z = -c + a. Substituting z into the other equations to obtain x and y, I ended up with the following solution:
x= -6a - 2b + 13c
y = b + 4a -5c
z = a - c
I was wondering if my method of solving is valid, and if the answer I obtained seems reasonable. Thanks.

 

[Student100]

Homework Statement

Find all solutions of the linear system
x + 2y + 3z = a
x + 3y + 8z = b
x + 2y + 2z = c
where a,b, and c are arbitrary constants.

Homework Equations

The Attempt at a Solution

Using elimination, I managed to set the coefficients on the diagonal equal to 1, which then allowed me to solve for z, which was z = -c + a. Substituting z into the other equations to obtain x and y, I ended up with the following solution:
x= -6a - 2b + 13c
y = b + 4a -5c
z = a - c
I was wondering if my method of solving is valid, and if the answer I obtained seems reasonable. Thanks.

Your z looks correct, but I get something different for y and x. Recheck your work, and I'll recheck mine.

 

[yango_17]

I plugged my values back into the original system and the equalities don't match up. Now I'm just trying to find out where my mistake was.

 

[Student100]

I plugged my values back into the original system and the equalities don't match up. Now I'm just trying to find out where my mistake was.

I just redid it, your Z is correct, so start back from there. I'm assuming you just made a calculation error with y. Your a is positive (and smaller than it should be), while it should be negative. Same deal with your C.

 

[Faithp]

Hey did you manage to solve this ...please help wit the proceedure

 

[WWGD]

Hey did you manage to solve this ...please help wit the proceedure

Write it as a matrix and do Gaussian elimination.

 

[Fred Wright]

If you know how to evaluate determinants, you can solve this problem quite easily using Cramer's rule. Express the coefficients of x,y,z as a determinant:$$\Delta =\begin {vmatrix}
1&2&3 \\
1&3&8\\
1&2&2\\
\end {vmatrix}=-1\\
\Delta_x =
\begin {vmatrix}
a&2&3 \\
b&3&8\\
c&2&2\\
\end {vmatrix}=-10a+2b+7c\\
\Delta_y=
\begin {vmatrix}
1&a&3 \\
1&b&8\\
1&c&2\\
\end {vmatrix}=-6a+b+5c\\
\Delta_z=
\begin {vmatrix}
1&2&a \\
1&3&b\\
1&2&c\\
\end {vmatrix}=-a+c\\
$$Cramer's rule:
$$
x =\frac{\Delta_x}{\Delta}=10a-2b-7c\\
y=\frac{\Delta_y}{\Delta}=-6a+b+5c\\
z=\frac{\Delta_z}{\Delta}=a-c
$$

 

[epenguin]

All unnecessarily effortful. You should be able to see how by the special features of this set of equations you can solve it like lightning!