Cramer's Rule in Linear Equations

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What is the first equation in the system of linear equations?

3x - y - z = -2

2x - y + z = 5

x + 2y - 3z = -9

2x + y + z = 7 (correct)

Which variable appears with a positive coefficient in the second equation?

x (correct)

y

z

None of the above

How many variables are in the given system of equations?

Five

Two

Four

Three (correct)

What is the right-hand side value of the first equation?

7 Signup and view all the answers

Which method is suggested for solving the system of equations?

Cramer's Rule Signup and view all the answers

What is the correct matrix representation of the coefficients from the system of equations?

[[2, 1, 1], [3, -1, -1], [1, 2, -3]] Signup and view all the answers

What is the value of the determinant of the coefficient matrix if calculated correctly?

12 Signup and view all the answers

Which equation represents the substitution for $x$ using Cramer's Rule?

x = \frac{det(A_x)}{det(A)} Signup and view all the answers

Using Cramer's Rule, which determinant should be replaced to solve for $y$?

Replace the second column with the constants Signup and view all the answers

What is the correct solution for the variable $z$ after applying Cramer's Rule?

-1 Signup and view all the answers

What is the first step in applying Cramer's Rule to the given system of equations?

Calculate the determinant of the coefficient matrix. Signup and view all the answers

What is the coefficient of $z$ in the first equation?

0 Signup and view all the answers

Which equation correctly represents the value of $x$ using Cramer's Rule?

Determinant of matrix formed by replacing x-column with constant terms over the determinant of coefficient matrix. Signup and view all the answers

If the determinant of the coefficient matrix is 0, what can be inferred about the system of equations?

The system has either no solution or infinitely many solutions. Signup and view all the answers

What is the value of the constant on the right-hand side of the second equation?

-2 Signup and view all the answers

What is the first step in applying Cramer's Rule to solve the system of equations?

Set up the determinant of the coefficient matrix. Signup and view all the answers

For the system of equations provided, what is the coefficient matrix formed by the coefficients of x, y, and z?

[[2, 1, 1], [3, -1, -1], [1, 2, -3]] Signup and view all the answers

If the determinant of the coefficient matrix is zero, what does that imply about the system of equations?

The system has infinitely many solutions. Signup and view all the answers

What is the correct way to set up the determinant for the variable x using Cramer's Rule?

Replace the first column of the coefficient matrix with the constant terms. Signup and view all the answers

After finding the final values for x, y, and z using Cramer's Rule, what does each value represent?

The intersection point of all three planes. Signup and view all the answers