4.4 Cramer’s Rule

You can use the determinant of a matrix to help you solve a system of equations. For two equations with two variables written in ax + by = c form, you can construct a matrix of the coefficients of the variables. For the system the coefficient matrix is a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2

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. The determinant D of the coefficient matrix is

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 Step 1 Write the equations in standard form. Use Cramer’s rule to solve each system of equations.

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.

Check It Out! Example 2 Use Cramer’s rule to solve. Step 1 Write the equations in standard form.

Check It Out! Example 2 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. Because D = 0 and one of the numerator determinants is equal to 0, the system is dependent and has infinitely many solutions.

Cramer’s rule can be expanded to cover 3  3 systems. If D ≠ 0, then the system has a unique solution. If D = 0 and no numerator is 0, then the system is inconsistent. If D = 0 and at least one numerator is 0, then the system may be inconsistent or dependent.

Example 4: Nutrition Application 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 FoodCalories Protein4 Carbohydrates4 Fat9

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

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

Check It Out! Example 4 What if...? A diet requires 3200 calories, 700 grams of food, and 70% of the Calories from carbohydrates and fat. How many grams of protein, carbohydrates, and fat does the diet include? The diet will include p grams of protein, c grams of carbohydrates, and f grams of fat. Calories per Gram FoodCalories Protein4 Carbohydrates4 Fat9

Equation for total Calories Total grams of food Calories from carbohydrates and fat, 70%(3200) = Use a calculator. Check It Out! Example 4 Continued 4p + 4c + 9f = 3200 p + c + f = 700 4c + 9f = 2240

Check It Out! Example 4 Continued The diet includes 240 grams of protein, 380 grams of carbohydrates, and 80 grams of fat. p = 240c = 380f = 80