# Use Cramer’s Rule to solve the system of linear equations. 20x+8y=11 12x-24y=21

Question
Equations
Use Cramer’s Rule to solve the system of linear equations.
20x+8y=11
12x-24y=21

2021-02-05
Step 1
Given system of equations is
20x+8y=11
12x-24y=21
Firstly, we evaluate the value of determinant of coefficient matrix
$$\displaystyle{D}={\left|\begin{array}{cc} {20}&{8}\\{12}&-{24}\end{array}\right|}={20}{\left(-{24}\right)}-{12}{\left({8}\right)}=-{576}$$
Since, the value of determinant of coefficient matrix is non-zero,
So, given system of equation is consistent.
Step 2
Now, we consider the following determinants.
$$\displaystyle{D}_{{1}}={\left|\begin{array}{cc} {11}&{8}\\{21}&-{24}\end{array}\right|}={11}{\left(-{24}\right)}-{21}{\left({8}\right)}=-{432}$$
$$\displaystyle{D}_{{2}}={\left|\begin{array}{cc} {20}&{11}\\{12}&{21}\end{array}\right|}={20}{\left({21}\right)}-{12}{\left({11}\right)}={288}$$
Solution of given system of equations is
$$\displaystyle{x}=\frac{{D}_{{1}}}{{D}},{y}=\frac{{D}_{{2}}}{{D}}$$
$$\displaystyle{x}=\frac{{-{432}}}{{-{576}}},{y}=\frac{{{288}}}{{-{576}}}$$
$$\displaystyle{x}=\frac{{3}}{{4}},{y}=-\frac{{1}}{{2}}$$
Step 3
Result:
$$\displaystyle{x}=\frac{{4}}{{3}}.{y}=-\frac{{1}}{{2}}$$

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