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

Example: Write the coefficient matrix for the system: 2x = 5 – 3y 5y + 2x = 1

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

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.

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).

Example 2B: Using Cramer’s Rule for Two Equations Use Cramer’s rule to solve each system of equations. Step 1 Write the equations in standard form.

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.

Cramer’s rule can be expanded to cover 3  3 systems. 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? Calories per Gram Food Calories Protein 4 Carbohydrates Fat 9 The diet will include p grams of protein, c grams of carbohydrates, and f grams of fat.

Example 4 Continued 4p + 4c + 9f = 3400 Equation for total Calories Total grams of food p + c + f = 680 Calories from carbohydrates, 4c = 1700 Use a calculator.

Example 4 Continued p = 119 c = 725 f = 136 The diet includes 119 grams of protein, 425 grams of carbohydrates, and 136 grams of fat.

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