18. Systems of Equations and Matrices

Determinants and Cramer's Rule

18. Systems of Equations and Matrices

Determinants and Cramer's Rule

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Multiple Choice

Write each equation in standard form and use Cramer's Rule to solve the system.
$y-9x=-3$
$-3x=4y-1$

$y=3,x=0$

$x=0,y=3$

$x=-\frac13,y=1$

$x=\frac13,y=0$

Verified step by step guidance

First, rewrite each equation in standard form, which is Ax + By = C. For the first equation, y - 9x = -3, rearrange it to 9x - y = 3.

For the second equation, -3x = 4y - 1, rearrange it to 3x + 4y = 1.

Now, you have the system of equations in standard form: 9x - y = 3 and 3x + 4y = 1.

To use Cramer's Rule, calculate the determinant of the coefficient matrix, D, which is |9 -1; 3 4|. The determinant D is calculated as (9)(4) - (-1)(3).

Next, calculate the determinants for Dx and Dy. For Dx, replace the x-coefficients with the constants from the right side of the equations: |3 -1; 1 4|. For Dy, replace the y-coefficients: |9 3; 3 1|. Solve for x and y using x = Dx/D and y = Dy/D.