11.10: Solving Linear Equations Using Mathcad
- Page ID
- 135018
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
\(\begin{pmatrix}
5 \cdot x + 2 \cdot y + z = 36\\
x + 7 \cdot y + 3 \cdot z = 63\\
2 \cdot x + 3 \cdot y + 8 \cdot z = 81
\end{pmatrix}
solve, \begin{pmatrix}
x\\
y\\
z
\end{pmatrix} \rightarrow \begin{pmatrix}
\frac{18}{5} & \frac{27}{5} & \frac{36}{5}
\end{pmatrix} = \begin{pmatrix}
3.6 & 5.4 & 7.2
\end{pmatrix}\)