Using Cramer’s Rule, how do you compute the values of x and y?
Understand the Problem
The question is asking for the application of Cramer's Rule to solve for the variables $x$ and $y$ in a system of linear equations. It involves showing the method of using determinants to find the values of these variables.
Answer
$$ x = \frac{c_1b_2 - c_2b_1}{a_1b_2 - a_2b_1}, \quad y = \frac{a_1c_2 - a_2c_1}{a_1b_2 - a_2b_1} $$
Answer for screen readers
The values for ( x ) and ( y ) are given by: $$ x = \frac{c_1b_2 - c_2b_1}{a_1b_2 - a_2b_1}, \quad y = \frac{a_1c_2 - a_2c_1}{a_1b_2 - a_2b_1} $$
Steps to Solve
- Write the system of equations in matrix form
To apply Cramer's Rule, we first express the system of equations in matrix form. For example, consider the equations: $$ a_1x + b_1y = c_1 $$ $$ a_2x + b_2y = c_2 $$ This can be represented as ( AX = B ), where $$ A = \begin{pmatrix} a_1 & b_1 \ a_2 & b_2 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \end{pmatrix}, \quad B = \begin{pmatrix} c_1 \ c_2 \end{pmatrix} $$
- Calculate the determinant of matrix A
Next, we calculate the determinant of the matrix ( A ), denoted as ( |A| ): $$ |A| = a_1b_2 - a_2b_1 $$
- Compute determinants for ( D_x ) and ( D_y )
Now we need to find the determinants for ( D_x ) and ( D_y ) to solve for ( x ) and ( y ). To calculate ( D_x ), replace the first column of ( A ) with ( B ): $$ D_x = \begin{vmatrix} c_1 & b_1 \ c_2 & b_2 \end{vmatrix} = c_1b_2 - c_2b_1 $$ For ( D_y ), replace the second column of ( A ) with ( B ): $$ D_y = \begin{vmatrix} a_1 & c_1 \ a_2 & c_2 \end{vmatrix} = a_1c_2 - a_2c_1 $$
- Apply Cramer's Rule
Finally, we apply Cramer's Rule to find ( x ) and ( y ): $$ x = \frac{D_x}{|A|} = \frac{c_1b_2 - c_2b_1}{a_1b_2 - a_2b_1} $$ $$ y = \frac{D_y}{|A|} = \frac{a_1c_2 - a_2c_1}{a_1b_2 - a_2b_1} $$
The values for ( x ) and ( y ) are given by: $$ x = \frac{c_1b_2 - c_2b_1}{a_1b_2 - a_2b_1}, \quad y = \frac{a_1c_2 - a_2c_1}{a_1b_2 - a_2b_1} $$
More Information
Cramer's Rule is a useful method for solving systems of linear equations using determinants. It is particularly effective for small systems (2x2 or 3x3) but can become cumbersome for larger systems.
Tips
- Not calculating the determinants correctly: Ensure that the proper elements are being used when calculating ( D_x ) and ( D_y ).
- Forgetting to check if ( |A| \neq 0 ): If the determinant of ( A ) is zero, the system does not have a unique solution, making Cramer’s Rule inapplicable.
- Neglecting to replace the correct columns: Make sure to replace the right column in the determinant for ( D_x ) and ( D_y ).
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