Solve the system using Cramer's Rule. $$ \begin{array}{l}5 x-6 y-3 z=16 \ -2 x-4 y+6 z=-24 \ 3 x+4 y-5 z=18\end{array} $$ Find the determinant $$ D $$ (denominator). $$ D=-80 $$ Find the determinant $$ D_{z} $$ associated with $$ x $$. $$ D_{z}=80 $$ Find the determinant $$ D_{y} $$ associated with $$ y $$. $$ D_{y}=80 $$ Find the determinant $$ D_{z} $$ associated with $$ z $$. $$ D_{z}=400 $$ The solution is (x,y,z)= Submit Question

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UpStudy AI · Accepted Answer

We are given the system
$$
\begin{aligned}
5x - 6y - 3z &= 16, \
-2x - 4y + 6z &= -24, \
3x + 4y - 5z &= 18.
\end{aligned}
$$

The coefficient matrix $$A$$ is
$$
A=\begin{pmatrix}
5 & -6 & -3 \
-2 & -4 & 6 \
3 & 4 & -5
\end{pmatrix}.
$$

### 1. Compute $$ D $$ (the determinant of $$A$$)
We have
$$
D = \begin{vmatrix}
5 & -6 & -3 \
-2 & -4 & 6 \
3 & 4 & -5
\end{vmatrix}.
$$
Using cofactor expansion along the first row:
$$
\begin{aligned}
D =&\; 5\begin{vmatrix} -4 & 6 \ 4 & -5 \end{vmatrix}
- (-6)\begin{vmatrix} -2 & 6 \ 3 & -5 \end{vmatrix}
- 3\begin{vmatrix} -2 & -4 \ 3 & 4 \end{vmatrix} \
=&\; 5\Big[(-4)(-5)-6\cdot4\Big]
+ 6\Big[(-2)(-5)-6\cdot3\Big]
- 3\Big[(-2)(4)-(-4)(3)\Big] \
=&\; 5\Big[20-24\Big]
+6\Big[10-18\Big]
-3\Big[(-8)-(-12)\Big] \
=&\; 5(-4) + 6(-8) -3(-8) \
=&\; -20 -48 +24 \[1mm]
=&\; -80.
\end{aligned}
$$
Thus, $$\; D = -80.$$

### 2. Compute $$ D_x $$ (replace the $$x$$-column with the right-hand side)

Replace the first column of $$A$$ with the constants:
$$
A_x=\begin{pmatrix}
16 & -6 & -3 \
-24 & -4 & 6 \
18 & 4 & -5
\end{pmatrix}.
$$
Then
$$
D_x = \begin{vmatrix}
16 & -6 & -3 \
-24 & -4 & 6 \
18 & 4 & -5
\end{vmatrix}.
$$
Expanding along the first row:
$$
\begin{aligned}
D_x =&\; 16\begin{vmatrix} -4 & 6 \ 4 & -5 \end{vmatrix}
- (-6)\begin{vmatrix} -24 & 6 \ 18 & -5 \end{vmatrix}
- 3\begin{vmatrix} -24 & -4 \ 18 & 4 \end{vmatrix} \
=&\; 16\Big[(-4)(-5)-6\cdot4\Big]
+6\Big[(-24)(-5)-6\cdot18\Big]
-3\Big[(-24)(4)-(-4)(18)\Big] \
=&\; 16\Big[20-24\Big]
+6\Big[120-108\Big]
-3\Big[(-96)-(-72)\Big] \
=&\; 16(-4) +6(12) -3(-24) \
=&\; -64 + 72 + 72 \[1mm]
=&\; 80.
\end{aligned}
$$
Thus, $$ x = \frac{D_x}{D} = \frac{80}{-80} = -1. $$

### 3. Compute $$ D_y $$ (replace the $$y$$-column with the right-hand side)

Replace the second column of $$A$$ with the constants:
$$
A_y=\begin{pmatrix}
5 & 16 & -3 \
-2 & -24 & 6 \
3 & 18 & -5
\end{pmatrix}.
$$
Then
$$
D_y = \begin{vmatrix}
5 & 16 & -3 \
-2 & -24 & 6 \
3 & 18 & -5
\end{vmatrix}.
$$
Expanding along the first row:
$$
\begin{aligned}
D_y =&\; 5\begin{vmatrix} -24 & 6 \ 18 & -5 \end{vmatrix}
- 16\begin{vmatrix} -2 & 6 \ 3 & -5 \end{vmatrix}
- 3\begin{vmatrix} -2 & -24 \ 3 & 18 \end{vmatrix} \
=&\; 5\Big[(-24)(-5)-6\cdot18\Big]
-16\Big[(-2)(-5)-6\cdot3\Big]
-3\Big[(-2)(18)-(-24)(3)\Big] \
=&\; 5\Big[120-108\Big]
-16\Big[10-18\Big]
-3\Big[(-36)-(-72)\Big] \
=&\; 5(12)
-16(-8)
-3(36) \
=&\; 60 + 128 - 108 \[1mm]
=&\; 80.
\end{aligned}
$$
Thus, $$ y = \frac{D_y}{D} = \frac{80}{-80} = -1. $$

### 4. Compute $$ D_z $$ (replace the $$z$$-column with the right-hand side)

Replace the third column of $$A$$ with the constants:
$$
A_z=\begin{pmatrix}
5 & -6 & 16 \
-2 & -4 & -24 \
3 & 4 & 18 
\end{pmatrix}.
$$
Then
$$
D_z = \begin{vmatrix}
5 & -6 & 16 \
-2 & -4 & -24 \
3 & 4 & 18 
\end{vmatrix}.
$$
Expanding along the first row:
$$
\begin{aligned}
D_z =&\; 5\begin{vmatrix} -4 & -24 \ 4 & 18 \end{vmatrix}
- (-6)\begin{vmatrix} -2 & -24 \ 3 & 18 \end{vmatrix}
+ 16\begin{vmatrix} -2 & -4 \ 3 & 4 \end{vmatrix} \
=&\; 5\Big[(-4)(18)-(-24)(4)\Big]
+6\Big[(-2)(18)-(-24)(3)\Big]
+16\Big[(-2)(4)-(-4)(3)\Big] \
=&\; 5\Big[-72 + 96\Big]
+6\Big[-36 + 72\Big]
+16\Big[-8+12\Big] \
=&\; 5(24) +6(36)+16(4) \
=&\; 120+216+64 \
=&\; 400.
\end{aligned}
$$
Thus, $$ z = \frac{D_z}{D} = \frac{400}{-80} = -5. $$

### Final Answer

The solution to the system is:
$$
(x,y,z)=(-1,-1,-5).
$$
The solution is $$ x = -1 $$, $$ y = -1 $$, and $$ z = -5 $$.

Solve the system using Cramer’s Rule.

Find the determinant (denominator).

Find the determinant associated with .

Find the determinant associated with .

Find the determinant associated with .

The solution is (x,y,z)=
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Solution

1. Compute (the determinant of )

2. Compute (replace the -column with the right-hand side)

3. Compute (replace the -column with the right-hand side)

4. Compute (replace the -column with the right-hand side)

Final Answer

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Solve the system using Cramer’s Rule. Find the determinant (denominator). Find the determinant associated with . Find the determinant associated with . Find the determinant associated with . The solution is (x,y,z)= Submit Question