To solve a system of two linear equations in two variables using Cramer's rule, follow this six-step procedure: 
1. Evaluate the determinant $$D$$, using the coefficients of the variables. 
2. Evaluate the determinant $$D_x$$ by using the system's constants in place of the $$x$$ coefficients. 
3. Evaluate the determinant $$D_y$$ by using the system's constants in place of the $$y$$ coefficients. 
4. Find the values of $$x$$ and $$y$$ using the formulas $$x = \frac{D_x}{D}$$ and $$y = \frac{D_y}{D}$$. 
5. Write the final solution as an ordered pair $$(x, y)$$. 
6. Check that the ordered pair is a valid solution by substituting it back into both original equations.

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Solving systems of linear equations can be achieved using a method called Cramer's rule. Preparing to use Cramer's rule requires learning new definitions and notation, specifically how to evaluate the determinants of $$2 	imes 2$$ and $$3 	imes 3$$ matrices, before applying the rule to solve systems of equations and application problems.

Solving Systems of Linear Equations Using Determinants

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Solving a System of Two Linear Equations Using Cramer's Rule

For a system of two linear equations in the standard form $$\left\{\begin{array}{l}a_1x+b_1y=k_1\\a_2x+b_2y=k_2\end{array}\right.$$, Cramer's Rule states that the solution $$(x, y)$$ can be calculated using determinants: $$x = \frac{D_x}{D}$$ and $$y = \frac{D_y}{D}$$. To use this method, you must first evaluate three specific determinants. The determinant $$D$$ is formed using only the coefficients of the variables: $$D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$$. The determinant $$D_x$$ is formed by substituting the constants $$k_1$$ and $$k_2$$ in place of the $$x$$ coefficients: $$D_x = \begin{vmatrix} k_1 & b_1 \\ k_2 & b_2 \end{vmatrix}$$. Similarly, the determinant $$D_y$$ is formed by substituting the constants in place of the $$y$$ coefficients: $$D_y = \begin{vmatrix} a_1 & k_1 \\ a_2 & k_2 \end{vmatrix}$$.

Cramer's Rule for a System of Two Linear Equations

Solve the system $$\left\{\begin{array}{l} 2x + y = -4 \\ 3x - 2y = -6 \end{array}\right.$$ using Cramer's rule. First, evaluate the determinant $$D$$ using the coefficients of the variables: $$D = \begin{vmatrix} 2 & 1 \\ 3 & -2 \end{vmatrix} = (2)(-2) - (1)(3) = -7$$. Next, evaluate $$D_x$$ by replacing the $$x$$ coefficients with the constants $$-4$$ and $$-6$$: $$D_x = \begin{vmatrix} -4 & 1 \\ -6 & -2 \end{vmatrix} = (-4)(-2) - (1)(-6) = 14$$. Then, evaluate $$D_y$$ by replacing the $$y$$ coefficients with the constants: $$D_y = \begin{vmatrix} 2 & -4 \\ 3 & -6 \end{vmatrix} = (2)(-6) - (-4)(3) = 0$$. Now, find $$x$$ and $$y$$ using the formulas: $$x = \frac{D_x}{D} = \frac{14}{-7} = -2$$ and $$y = \frac{D_y}{D} = \frac{0}{-7} = 0$$. Write the solution as the ordered pair $$(-2, 0)$$. Finally, check the solution in both original equations to verify it is correct.

Example: Solving a System of Two Equations Using Cramer's Rule

Apply Cramer's rule to solve a system of two linear equations, such as $$\left\{\begin{array}{l} 3x + y = -3 \\ 2x + 3y = 6 \end{array}\right.$$. First, evaluate the main determinant $$D$$ using the coefficients of the variables. Next, evaluate the determinants $$D_x$$ and $$D_y$$ by substituting the system's constants in place of the respective variable coefficients. Then, calculate the values of the variables using the formulas $$x = \frac{D_x}{D}$$ and $$y = \frac{D_y}{D}$$. Finally, state the solution as an ordered pair and verify it by substituting the values back into the original equations.

Practice: Solving a System of Two Equations Using Cramer's Rule

For a system of three linear equations represented as:
$$\left\{\begin{array}{l} a_1 x + b_1 y + c_1 z = k_1 \\\\ a_2 x + b_2 y + c_2 z = k_2 \\\\ a_3 x + b_3 y + c_3 z = k_3 \end{array}\right.$$

The solution $$(x, y, z)$$ can be determined using determinants:
$$x = \frac{D_x}{D}$$, $$y = \frac{D_y}{D}$$, and $$z = \frac{D_z}{D}$$

Where $$D$$ is the determinant of the coefficients of the variables:
$$D = \begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \end{vmatrix}$$

$$D_x$$ is formed by replacing the $$x$$-coefficients with the constants $$k_1$$, $$k_2$$, and $$k_3$$:
$$D_x = \begin{vmatrix} k_1 & b_1 & c_1 \\\\ k_2 & b_2 & c_2 \\\\ k_3 & b_3 & c_3 \end{vmatrix}$$

$$D_y$$ is formed by replacing the $$y$$-coefficients with the constants:
$$D_y = \begin{vmatrix} a_1 & k_1 & c_1 \\\\ a_2 & k_2 & c_2 \\\\ a_3 & k_3 & c_3 \end{vmatrix}$$

$$D_z$$ is formed by replacing the $$z$$-coefficients with the constants:
$$D_z = \begin{vmatrix} a_1 & b_1 & k_1 \\\\ a_2 & b_2 & k_2 \\\\ a_3 & b_3 & k_3 \end{vmatrix}$$

Cramer's Rule for a System of Three Linear Equations

Three points $$(x_1, y_1)$$, $$(x_2, y_2)$$, and $$(x_3, y_3)$$ are collinear if and only if the determinant of the matrix formed by their coordinates and a column of ones is equal to zero:

$$\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0$$

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