# 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

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