College Algebra (Math 1314): Test 4 Review (9.3, 2.6, and 6.5)

Front

What is the determinant of a 2x2 matrix?

Back

For a matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the determinant is calculated as \( ad - bc \).

Front

How do you isolate a radical in an equation?

Back

To isolate a radical, you can add or subtract terms from both sides of the equation, ensuring the radical is alone on one side.

Front

What is Cramer's Rule?

Back

Cramer's Rule is a mathematical theorem used to solve systems of linear equations using determinants.

Front

How do you apply Cramer's Rule to solve for variables in a system of equations?

Back

1. Write the system in matrix form. 2. Calculate the determinant of the coefficient matrix. 3. Replace one column with the constants and calculate the determinant. 4. Divide the determinant of the modified matrix by the determinant of the coefficient matrix.

Front

What is the solution to the system of equations 3x + 2y = 16 and 7x + y = 19 using Cramer's Rule?

Back

The solution is (2, 5).

Front

What does it mean if the determinant of a matrix is zero?

Back

If the determinant is zero, the matrix is singular, meaning it does not have an inverse and the system of equations may have no solution or infinitely many solutions.

Front

How do you solve for x in the equation \( \sqrt[3]{x} = 10 \)?

Back

To solve for x, cube both sides: \( x = 10^3 = 1000 \).

Front

What is the formula for the determinant of a 3x3 matrix?

Back

For a matrix \( \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \), the determinant is \( a(ei - fh) - b(di - fg) + c(dh - eg) \).

Front

How do you check if a solution to a system of equations is correct?

Back

Substitute the values of the variables back into the original equations to see if they satisfy all equations.

Front

What is the significance of the coefficients in a linear equation?

Back

The coefficients represent the rate of change of the dependent variable with respect to the independent variable.