Holt Algebra 2

4-4 Determinants and Cramer’s Rule

The coefficient matrix for a system of linear equations in standard form is the matrix formed by the coefficients for the variables in the equations.

For the system

the coefficient matrix is

a1x + b1y = c1

a2x + b2y = c2

Holt Algebra 2

4-4 Determinants and Cramer’s Rule

Example:Write the coefficient matrix for the system:

2x = 5 – 3y5y + 2x = 1

Holt Algebra 2

4-4 Determinants and Cramer’s Rule

Holt Algebra 2

4-4 Determinants and Cramer’s Rule

You can use Cramer’s rule to tell whether the system represented by the matrix has one solution, no solution, or infinitely many solutions.

Holt Algebra 2

4-4 Determinants and Cramer’s Rule

Example 2A: Using Cramer’s Rule for Two Equations

Use Cramer’s rule to solve each system of equations.

Step 1 Find D, the determinant of the coefficient matrix.

D 0, so the system is consistent.

Holt Algebra 2

4-4 Determinants and Cramer’s Rule

Example 2A Continued

Step 2 Solve for each variable by replacing the coefficients of that variable with the constants as shown below.

The solution is (4, 2).

Holt Algebra 2

4-4 Determinants and Cramer’s Rule

Example 2B: Using Cramer’s Rule for Two Equations

Step 1 Write the equations in standard form.

Use Cramer’s rule to solve each system of equations.

Holt Algebra 2

4-4 Determinants and Cramer’s Rule

Example 2B Continued

Step 2 Find the determinant of the coefficient matrix.

D = 0, so the system is either inconsistent or dependent. Check the numerators for x and y to see if either is 0.

Since at least one numerator is 0, the system is dependent and has infinitely many solutions.

Holt Algebra 2

4-4 Determinants and Cramer’s Rule

A nutritionist creates a diet for a long-distance runner that includes 3400 Calories from 680 grams of food, with half the Calories coming from carbohydrates. How many grams of protein, carbohydrates, and fat will this diet include?

The diet will include p grams of protein, c grams of carbohydrates, and f grams of fat.

Calories per Gram

Food Calories

Protein 4

Carbohydrates 4

Fat 9

Cramer’s rule can be expanded to cover 3 3 systems.

Holt Algebra 2

4-4 Determinants and Cramer’s Rule

Example 4 Continued

Equation for total Calories

Total grams of food

Use a calculator.

4p + 4c + 9f = 3400

4c = 1700

p + c + f = 680

Calories from carbohydrates,

Holt Algebra 2

4-4 Determinants and Cramer’s Rule

The diet includes 119 grams of protein, 425 grams of carbohydrates, and 136 grams of fat.

Example 4 Continued

p = 119 c = 725 f = 136

Holt Algebra 2

4-4 Determinants and Cramer’s Rule

HW pg. 275 # 19, 20, 29, 33,40, 43