4.8 Solving Simultaneous Linear Equations using Matrices

4.8 Solving Simultaneous Linear Equations using Matrices
1. Two simultaneous linear equations can be written in the matrix equation form.
For example, in the simultaneous equations:
ax + by = e
cx + dy = f
can be written in the matrix form as follows:

$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}e\\ f\end{array}\right),$

Where a, b, c, d, e and are constant while x and y are unknowns.

Example 1:
Write the following simultaneous linear equations in the matrix form.
y– 6x – 19 = 0
2y + 3x + 22 = 0

Solution:
– 6x + y = 19
3x + 2y = – 22
The matrix form is:
$\left(\begin{array}{cc}-6& 1\\ 3& 2\end{array}\right)\left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}19\\ -22\end{array}\right)$

2. Matrix equations in the form $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}e\\ f\end{array}\right)$
can be solved for the unknowns x and as follows.

(a) Let $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ and find A-1.

(b) Multiply both sides of the equation by A-1.

${A}^{-1}\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{l}x\\ y\end{array}\right)={A}^{-1}\left(\begin{array}{l}e\\ f\end{array}\right)$

Example 2:
Solve the following simultaneous equations by using the matrix method.
2x = 5 – 3y
7x = 1 – 5y

Solution:
2x + 3y = 5
7x + 5y = 1
$\begin{array}{l}\left(\begin{array}{cc}2& 3\\ 7& 5\end{array}\right)\left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}5\\ 1\end{array}\right)←\overline{)\begin{array}{l}\text{write the simultaneous}\\ \text{equations in matrix form}\text{.}\end{array}}\\ \text{Let}A=\left(\begin{array}{cc}2& 3\\ 7& 5\end{array}\right)\\ {A}^{-1}=\frac{1}{ad-bc}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)\\ {A}^{-1}=\frac{1}{10-21}\left(\begin{array}{cc}5& -3\\ -7& 2\end{array}\right)\\ {A}^{-1}=\frac{1}{-11}\left(\begin{array}{cc}5& -3\\ -7& 2\end{array}\right)\\ \\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{-11}\left(\begin{array}{cc}5& -3\\ -7& 2\end{array}\right)\left(\begin{array}{l}5\\ 1\end{array}\right)←\overline{)\left(\begin{array}{l}x\\ y\end{array}\right)={A}^{-1}\left(\begin{array}{l}e\\ f\end{array}\right)}\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{-11}\left(\begin{array}{l}5×5+\left(-3\right)×1\\ -7×5+2×1\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\frac{1}{-11}\left(\begin{array}{l}22\\ -33\end{array}\right)\\ \left(\begin{array}{l}x\\ y\end{array}\right)=\left(\begin{array}{l}-\frac{22}{11}\\ \frac{-33}{-11}\end{array}\right)=\left(\begin{array}{l}-2\\ 3\end{array}\right)\\ \therefore x=-2,\text{}y=3.\end{array}$

3 Comments to “4.8 Solving Simultaneous Linear Equations using Matrices”

1. Gurdit Singh

Hi your explanation of simultaneous equation where a, b, c, d, e and f is wrong. According to your example, the numbers don’t fall into the places you explained for a,b,c,d and e.

1. myhometuition

Dear Gurdit Singh,
thanks for pointing out our mistake, correction had been made accordingly.

2. This is nice!