Solve the system using Cramer's Rule. $$ \begin{array}{l} 4 x+2 y=14 \ 3 x-y=18 \end{array} $$ Find the determinant $$ D $$ (denominator). $$ D= $$ $$ \square $$ Find the determinant $$ D_{x} $$ associated with $$ x $$. $$ D_{z}=\square $$ Find the determinant $$ D_{y} $$ associated with $$ y $$. $$ D_{v}=\square $$ The solution is $$ (x, y)=1 $$ $$ \square $$ , $$ \square $$ ). Question Help: Video

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**Step 1. Compute the determinant $$ D $$ of the coefficient matrix.**

The system is:
$$
\begin{aligned}
4x + 2y &= 14, \
3x - y &= 18.
\end{aligned}
$$
The coefficient matrix is:
$$
\begin{pmatrix}
4 & 2 \
3 & -1
\end{pmatrix}.
$$
The determinant is computed as:
$$
D = 4(-1) - 2(3) = -4 - 6 = -10.
$$

**Step 2. Compute the determinant $$ D_x $$ (associated with $$ x $$).**

Replace the $$ x $$-column in the coefficient matrix with the constants:
$$
\begin{pmatrix}
14 & 2 \
18 & -1
\end{pmatrix}.
$$
Then,
$$
D_x = 14(-1) - 2(18) = -14 - 36 = -50.
$$

**Step 3. Compute the determinant $$ D_y $$ (associated with $$ y $$).**

Replace the $$ y $$-column in the coefficient matrix with the constants:
$$
\begin{pmatrix}
4 & 14 \
3 & 18
\end{pmatrix}.
$$
Then,
$$
D_y = 4(18) - 14(3) = 72 - 42 = 30.
$$

**Step 4. Find the solution using Cramer’s Rule.**

Cramer’s Rule gives:
$$
x = \frac{D_x}{D} = \frac{-50}{-10} = 5,
$$
$$
y = \frac{D_y}{D} = \frac{30}{-10} = -3.
$$

**Final Answer:**
$$
D = -10, \quad D_x = -50, \quad D_y = 30, \quad (x, y) = (5, -3).
$$
**Determinants and Solution:** - **Determinant $$ D $$:** -10 - **Determinant $$ D_x $$:** -50 - **Determinant $$ D_y $$:** 30 - **Solution:** $$ (x, y) = (5, -3) $$

Solve the system using Cramer’s Rule.

Find the determinant (denominator).

Find the determinant associated with .

Find the determinant associated with .

The solution is , ).
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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 . The solution is , ). Question Help: Video