Cramer's Rule in Linear Equations

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Questions and Answers

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?

<p>7 (D)</p> Signup and view all the answers

Which method is suggested for solving the system of equations?

<p>Cramer's Rule (A)</p> Signup and view all the answers

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

<p>[[2, 1, 1], [3, -1, -1], [1, 2, -3]] (A)</p> Signup and view all the answers

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

<p>12 (A)</p> Signup and view all the answers

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

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

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

<p>Replace the second column with the constants (C)</p> Signup and view all the answers

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

<p>-1 (C)</p> Signup and view all the answers

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

<p>Calculate the determinant of the coefficient matrix. (A)</p> Signup and view all the answers

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

<p>0 (D)</p> Signup and view all the answers

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

<p>Determinant of matrix formed by replacing x-column with constant terms over the determinant of coefficient matrix. (C)</p> Signup and view all the answers

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

<p>The system has either no solution or infinitely many solutions. (B)</p> Signup and view all the answers

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

<p>-2 (C)</p> Signup and view all the answers

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

<p>Set up the determinant of the coefficient matrix. (B)</p> 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?

<p>[[2, 1, 1], [3, -1, -1], [1, 2, -3]] (C)</p> Signup and view all the answers

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

<p>The system has infinitely many solutions. (A)</p> Signup and view all the answers

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

<p>Replace the first column of the coefficient matrix with the constant terms. (C)</p> 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?

<p>The intersection point of all three planes. (B)</p> Signup and view all the answers

Flashcards

System of equations

A set of two or more equations with multiple variables.

Cramer's Rule

A method for solving systems of linear equations using determinants.

Three Linear Equations

Three equations, each with three variables(e.g. x, y, z).

Variables

Symbols representing unknown values.

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Solving for x, y, z

Finding the specific numerical values for x, y, and z.

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Determinant of a Matrix

A scalar value associated with a square matrix. It's calculated using a specific formula and can be used to solve systems of equations.

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Coefficient Matrix

A matrix formed by the coefficients of the variables in a system of equations.

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Solve for x using Cramer's Rule

To find the value of x in a system of equations, you calculate the determinant of the coefficient matrix, replace the x-column with the constant terms, and then divide the new determinant by the original determinant.

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Solve for y using Cramer's Rule

To find the value of y in a system of equations, you calculate the determinant of the coefficient matrix, replace the y-column with the constant terms, and then divide the new determinant by the original determinant.

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Solve for z using Cramer's Rule

To find the value of z in a system of equations, you calculate the determinant of the coefficient matrix, replace the z-column with the constant terms, and then divide the new determinant by the original determinant.

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Determinant of the system

The determinant calculated from the coefficients of the variables in the system of equations.

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Determinant of x

The determinant calculated by replacing the x-coefficient column with the constant terms.

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Determinant of y

The determinant calculated by replacing the y-coefficient column with the constant terms.

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Determinant of z

The determinant calculated by replacing the z-coefficient column with the constant terms.

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Cramer's Rule Formula

The solution for each variable (x, y, z) is found by dividing the determinant of that variable by the determinant of the system.

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Constant Terms Matrix

A matrix formed by the constant terms on the right side of the equation. Used in Cramer's Rule to replace columns.

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Replace the x-column

Replace the first column of the coefficient matrix with the constant terms matrix to calculate the determinant.

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Replace the y-column

Replace the second column of the coefficient matrix with the constant terms matrix to calculate the determinant.

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Replace the z-column

Replace the third column of the coefficient matrix with the constant terms matrix to calculate the determinant.

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