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how to answer this system of equations using cramer's rule

How To Answer This System Of Equations Using Cramers Rule class=

Sagot :

Given:

Given the system of equations:

[tex]\begin{gathered} c+w+p=456 \\ c-p=80 \\ p=2w-2 \end{gathered}[/tex]

Required: Solution of the system using Cramer's rule

Explanation:

The system of equations can be rewritten as

[tex]\begin{gathered} c+p+w=456 \\ c-p+0w=80 \\ 0c+p-2w=-2 \end{gathered}[/tex]

Write down the augmented matrix.

[tex]\begin{bmatrix}{1} & {1} & {1} & {456} \\ {1} & {-1} & {0} & {80} \\ {0} & {1} & {-2} & {-2} \\ {} & {} & {} & {}\end{bmatrix}[/tex]

Calculate the main determinant.

[tex]\begin{gathered} D=\det\begin{bmatrix}{1} & {1} & {1} \\ {1} & {-1} & {0} \\ {0} & {1} & {-2}\end{bmatrix} \\ =1\left(2-0\right)-1\left(-2-1\right) \\ =2+3 \\ =5 \end{gathered}[/tex]

Substitute the c-column with RHS and find the determinant.

[tex]\begin{gathered} D_c=\det\begin{bmatrix}{456} & {1} & {1} \\ {80} & {-1} & {0} \\ {-2} & {1} & {-2}\end{bmatrix} \\ =456\left(2-0\right)-80\left(-2-1\right)-2\left(0+1\right) \\ =912+240-2=1150 \end{gathered}[/tex]

Substitute the p-column with RHS and find the determinant.

[tex]\begin{gathered} D_p=\det\begin{bmatrix}{1} & {456} & {1} \\ {1} & {80} & {0} \\ {0} & {-2} & {-2}\end{bmatrix} \\ =1(-160-0)-1(-912+2) \\ =-160+910 \\ =750 \end{gathered}[/tex]

Substitute the w-column with RHS and find the determinant.

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