Use Cramer’s Rule to solve (if possible) the system of linear equations.

4x-y-z=1

2x+2y+3z=10

5x-2y-2z=-1

4x-y-z=1

2x+2y+3z=10

5x-2y-2z=-1

Wribreeminsl
2021-02-05
Answered

4x-y-z=1

2x+2y+3z=10

5x-2y-2z=-1

You can still ask an expert for help

Margot Mill

Answered 2021-02-06
Author has **106** answers

Step 1

Given system of equations are

4x-y-z=1

2x+2y+3z=10

5x-2y-2z=-1

Firstly, we will find determinant of coefficient matrix.

$D=\left[\begin{array}{ccc}4& -1& -1\\ 2& 2& 3\\ 5& -2& -2\end{array}\right]=4\left|\begin{array}{cc}2& 3\\ -2& -2\end{array}\right|-(-1)\left|\begin{array}{cc}2& 3\\ 5& -2\end{array}\right|-1\left|\begin{array}{cc}2& 2\\ 5& -2\end{array}\right|$

D=4(-4+6)+(-4-15)-(-4-10)=8-19+14=3

Step 2

Now, we will find the following determinants.

${D}_{1}=\left[\begin{array}{ccc}1& -1& -1\\ 10& 2& 3\\ -1& -2& -2\end{array}\right]=1\left|\begin{array}{cc}2& 3\\ -2& -2\end{array}\right|-(-1)\left|\begin{array}{cc}10& 3\\ -1& -2\end{array}\right|-1\left|\begin{array}{cc}10& 2\\ -1& -2\end{array}\right|$

${D}_{1}=(-4+6)+(-20+3)-(-20+2)=2-17+18=3$

${D}_{2}=\left[\begin{array}{ccc}4& 1& -1\\ 2& 10& 3\\ 5& -1& -2\end{array}\right]=4\left|\begin{array}{cc}10& 3\\ -1& -2\end{array}\right|-1\left|\begin{array}{cc}2& 3\\ 5& -2\end{array}\right|-1\left|\begin{array}{cc}2& 10\\ 5& -1\end{array}\right|$

${D}_{2}=4(-20+3)-(-4-15)-(-2-50)=-68+19+52=3$

${D}_{3}=\left[\begin{array}{ccc}4& -1& 1\\ 2& 2& 10\\ 5& -2& -1\end{array}\right]=4\left|\begin{array}{cc}2& 10\\ -2& -1\end{array}\right|-(-1)\left|\begin{array}{cc}2& 10\\ 5& -1\end{array}\right|+1\left|\begin{array}{cc}2& 2\\ 5& -2\end{array}\right|$

${D}_{3}=4(-2+20)+(-2-50)+(-4-10)=6$

Now, solution of given system of equations is

$x=\frac{{D}_{1}}{D},y=\frac{{D}_{2}}{D},z=\frac{{D}_{3}}{D}$

x=1,y=1,z=2

Step 3

Result: Solution of given system of equations is

x=1, y=1, z=2

Given system of equations are

4x-y-z=1

2x+2y+3z=10

5x-2y-2z=-1

Firstly, we will find determinant of coefficient matrix.

D=4(-4+6)+(-4-15)-(-4-10)=8-19+14=3

Step 2

Now, we will find the following determinants.

Now, solution of given system of equations is

x=1,y=1,z=2

Step 3

Result: Solution of given system of equations is

x=1, y=1, z=2

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