Solve the system of equations using determinants.
-3x + 2y = 26 3x - 3y = 6
A) (-4, 6)
B) (-6, -4)
C) (6, 4)
D) (4, 6)

Question

Quizplus · Accepted Answer

Using determinants, we can write the system of equations in matrix form:
$$\begin{pmatrix}3 & 2\3 & -3\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}26\6\end{pmatrix}$$
The determinant of the coefficient matrix is:
$$\begin{vmatrix}3 & 2\3 & -3\end{vmatrix}=(3)(-3)-(2)(3)=-9-6=-15$$
The determinant of the matrix obtained by replacing the first column of the coefficient matrix with the constant terms is:
$$\begin{vmatrix}26 & 2\6 & -3\end{vmatrix}=(26)(-3)-(2)(6)=-78-12=-90$$
The determinant of the matrix obtained by replacing the second column of the coefficient matrix with the constant terms is:
$$\begin{vmatrix}3 & 26\3 & 6\end{vmatrix}=(3)(6)-(26)(3)=-72$$
By Cramer's rule, the solution is:
$$x=\frac{\begin{vmatrix}26 & 2\6 & -3\end{vmatrix}}{\begin{vmatrix}3 & 2\3 & -3\end{vmatrix}}=\frac{-90}{-15}=6$$
$$y=\frac{\begin{vmatrix}3 & 26\3 & 6\end{vmatrix}}{\begin{vmatrix}3 & 2\3 & -3\end{vmatrix}}=\frac{-72}{-15}=4$$
Therefore, the solution is $(x,y)=(6,4)$, which corresponds to choice C. We can use determinants to solve this system of equations:

First, we write the system in matrix form:
\begin{pmatrix}3 & 2 \3 & -3\end{pmatrix} \begin{pmatrix}x \y\end{pmatrix} = \begin{pmatrix}26 \6\end{pmatrix}

Then, we can find the determinant of the coefficient matrix:
\begin{vmatrix}3 & 2 \3 & -3\end{vmatrix} = (3)(-3) - (2)(3) = -9 - 6 = -15

Next, we replace the first column of the coefficient matrix with the constants vector:
\begin{pmatrix}26 & 2 \6 & -3\end{pmatrix}

And find the determinant of this matrix:
\begin{vmatrix}26 & 2 \6 & -3\end{vmatrix} = (26)(-3) - (2)(6) = -78 - 12 = -90

Finally, we replace the second column of the coefficient matrix with the constants vector:
\begin{pmatrix}3 & 26 \3 & 6\end{pmatrix}

And find the determinant of this matrix:
\begin{vmatrix}3 & 26 \3 & 6\end{vmatrix} = (3)(6) - (26)(3) = 18 - 78 = -60

Using the formulas for Cramer's Rule, we can find the values of x and y:
x = \frac{\begin{vmatrix}26 & 2 \6 & -3\end{vmatrix}}{\begin{vmatrix}3 & 2 \3 & -3\end{vmatrix}} = \frac{-90}{-15} = 6
y = \frac{\begin{vmatrix}3 & 26 \3 & 6\end{vmatrix}}{\begin{vmatrix}3 & 2 \3 & -3\end{vmatrix}} = \frac{-60}{-15} = 4

Therefore, the solution to the system of equations is (6,4), which corresponds to choice C.

Solve the System of Equations Using Determinants