# Use Cramer's rule to solve the following system of equations. If D = 0, use another method to solve the system. 2x + 3y = 4 4x - 6y = 8

Question
Equations
Use Cramer's rule to solve the following system of equations. If D = 0, use another method to solve the system.
2x + 3y = 4
4x - 6y = 8

2020-12-01
Step 1
Given: The system of equations is
2x+3y=4
4x-6y=8
Step 2
Solution:
Obtain the value of D as follows.
$$\displaystyle{D}{\left|\begin{array}{cc} {2}&{3}\\{4}&-{6}\end{array}\right|}$$
=2(-6)-(3)(4)
=-12-12
$$\displaystyle=-{24}\ne{0}$$
As D is not equal to zero, the unique solution exists.
Step 3
Obtain the value of $$\displaystyle{D}_{{1}}$$ as follows.
$$\displaystyle{D}_{{1}}={\left|\begin{array}{cc} {4}&{3}\\{8}&-{6}\end{array}\right|}$$
=(4)(-6)-3(8)
=-24-24
=-48
Step 4
Obtain the value of $$\displaystyle{D}_{{2}}$$ as follows.
$$\displaystyle{D}_{{2}}={\left|\begin{array}{cc} {2}&{4}\\{4}&{8}\end{array}\right|}$$
=2(8)-(4)(4)
=16-16
=0
Step 5
Thus, the value of x and y by Cramer’s rule becomes
$$\displaystyle{x}=\frac{{D}_{{1}}}{{D}}=\frac{{-{48}}}{{-{24}}}={2}$$
$$\displaystyle{y}=\frac{{D}_{{2}}}{{D}}=\frac{{0}}{{-{24}}}={0}$$
Step 6
Therefore, the solution of the system of equations is
x=2 and y =0

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