Transcript 4-4 Cramer`s Rule

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
a1x + b1y = c1
a2x + b2y = c2
the coefficient matrix is
Holt Algebra 2
4-4 Determinants and Cramer’s Rule
Example:
Write the coefficient matrix for the system:
2x = 5 – 3y
5y + 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
Use Cramer’s rule to solve each system of
equations.
Step 1 Write the equations in standard form.
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
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
Protein
Carbohydrates
Fat
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Calories
4
4
9
The diet will include
p grams of protein,
c grams of
carbohydrates, and
f grams of fat.
4-4 Determinants and Cramer’s Rule
Example 4 Continued
4p + 4c + 9f = 3400
p + c + f = 680
4c = 1700
Use a calculator.
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Equation for total Calories
Total grams of food
Calories from carbohydrates,
4-4 Determinants and Cramer’s Rule
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.
Holt Algebra 2
4-4 Determinants and Cramer’s Rule
HW pg. 275
# 19, 20, 29, 33,40, 43
Holt Algebra 2