Question

Solve the following simultaneous equation using Cramer’s rule
$4m + 6n = 54:3m + 2n = 28$

Answer 1

To solve this type of question by Cramer’s rule firstly we need to know about Cramer’s Rule.
As per this rule, Cramer’s rule only works on square matrices that have a non-zero determinant and a unique solution.
Cramer’s rule is efficient for solving small systems and can be quickly calculated, however, as the system grows, calculating the new determinants can be very difficult and lengthy.
Firstly, to solve the variables is to find the determinant of coefficient of variables $m$ and $n$
Then find ${D_m}$and ${D_n}$ . In the final step divide the values of ${D_m}$and ${D_n}$with $D$ to find $m$and $n$ respectively, which in mathematical form is written as $m = \dfrac{{{D_m}}}{D},n = \dfrac{{{D_n}}}{D}$.

Complete answer:
The given question asks to find the value of $m$and$n$ for two equations which are
$4m + 6n = 54$
$3m + 2n = 28$
by using Cramer’s rule.
Cramer’s rule is represented as
$m = \dfrac{{{D_m}}}{D},n = \dfrac{{{D_n}}}{D}$
Where ${D_m},{D_n}$and $D$are the determinants.
Now, solve the given equation by calculating the two determinants. At first, we will calculate the value of$D$.
$D$= $4$ $6$ $3$ $2$
D$$$$ = 4 \times 2 - 6 \times 3
D$$$$ = 8 - 18
$D$ $= - 10$
Now we will find the value of ${D_m}$ which means that the value of $m$ term values in the first column will be replaced by the constant term after the equal sign, leaving the $n$ term value unchanged, by this we get,
{D_m} = $$$$54 $6$ $28$ $2$
${D_m} = 54 \times 2 - 6 \times 28$
${D_m} = 108 - 168$
${D_m} = - 60$
Similarly, we find the value ${D_n}$ which means that the value of $n$ term values in the first column will be replaced by the constant term after the equal sign, leaving the $m$ term value unchanged. By this we get,
{D_n} = $$$$4 $54$ $3$ $28$
${D_n} = 4 \times 28 - 54 \times 3$
${D_n} = 112 - 162$
${D_n} = - 50$
In the final step we will find $m$and $n$ values using the formula $m = \dfrac{{{D_m}}}{D},n = \dfrac{{{D_n}}}{D}$ we get:
Firstly, find the value of $m$, substituting the values we get,
$m = \dfrac{{{D_m}}}{D}$
$m = \dfrac{{ - 60}}{{ - 10}}$
$m = 6$
Now find the value of $n$, substitute the value we get,
$n = \dfrac{{{D_n}}}{D}$
$n = \dfrac{{ - 50}}{{ - 10}}$
$n = 5$
So, the values of $m$, $n$ are$(6,5)$ respectively for the equation by using Cramer’s rule.

Note:
To solve this type of question, one must always know how to expand the determinants and also one must know the concept of Cramer’s rule.
Also, this rule works for any number of variables on simultaneous equations.
Always be careful while solving the values of determinants.