468: Solving Linear Equations Using Mathcad

Numeric Methods: A system of equations is solved numerically using a Given/Find solve block. Mathcad requires seed values for each of the variables in the numeric method.

Seed values: x :=1 y :=1 z :=1

Given: \(5 \cdot x + 2 \cdot y + z = 36\) \(x + 7 \cdot y + 3 \cdot z = 63\) \(2 \cdot x + 3 \cdot y + 8 \cdot z = 81\)

Find (x, y, z) = \( \begin{pmatrix}
3.6\\
5.4\\
7.2
\end{pmatrix}\)

Other Given/Find solve blocks can be used.

Given \(\begin{pmatrix}
5 \cdot x + 2 \cdot y + z = 36\\
x + 7 \cdot y + 3 \cdot z = 63\\
2 \cdot x + 3 \cdot y + 8 \cdot z = 81
\end{pmatrix}
= \begin{pmatrix}
36\\
63\\
81
\end{pmatrix}\) Find(x, y, z) = \( \begin{pmatrix}
3.6\\
5.4\\
7.2
\end{pmatrix}\)

Given \(\begin{pmatrix}
5 & 2 & 1\\
1 & 7 & 3\\
2 & 3 & 8
\end{pmatrix} \cdot \begin{pmatrix}
x\\
y\\
z
\end{pmatrix} = \begin{pmatrix}
36\\
63\\
81
\end{pmatrix}\) Find(x, y, z) = \( \begin{pmatrix}
3.6\\
5.4\\
7.2
\end{pmatrix}\)

Matrix methods: The equations can also be solved using matrix algebra as shown below. In matrix form, the equations are written as MX = C. The solution vector is found by matrix mutiplication of by the inverse of M.

M:= \(\begin{pmatrix}
5 & 2 & 1\\
1 & 7 & 3\\
2 & 3 & 8
\end{pmatrix}\) C:= \(\begin{pmatrix}
36\\
63\\
81
\end{pmatrix}\) X := M^-1 \(\cdot\) C X = \(\begin{pmatrix}
3.6\\
5.4\\
7.2
\end{pmatrix}\)

Confirm that a solution has been found:

M \( \cdot\) X = \( \begin{pmatrix}
36\\
63\\
81
\end{pmatrix}\)

Alternative matrix solution using the lsolve command.

X := lsolve(M,C) X = \( \begin{pmatrix}
3.6\\
5.4\\
7.2
\end{pmatrix}\) M \( \cdot\) X = \( \begin{pmatrix}
36\\
63\\
81
\end{pmatrix}\)

Live symbolic method: To use the live symbolic method within this Mathcad document recursive definitions are required clear previous values of x, y and z. This would not be necessary if x, y and z had not been previous defined.

x := x y := y z := z

\(\begin{pmatrix}
5 \cdot x + 2 \cdot y + z = 36\\
x + 7 \cdot y + 3 \cdot z = 63\\
2 \cdot x + 3 \cdot y + 8 \cdot z = 81
\end{pmatrix}
solve, \begin{pmatrix}
x\\
y\\
z
\end{pmatrix} \rightarrow \begin{pmatrix}
\frac{18}{5} & \frac{27}{5} & \frac{36}{5}
\end{pmatrix} = \begin{pmatrix}
3.6 & 5.4 & 7.2
\end{pmatrix}\)