Question 151348
Problem: A survey of 615 teenagers found that 44% of the boys and 35% of the girls would like to be taller. Altogether, 231 teenagers in the survey wished they were taller. 
(1) Set up a system of equations to model this scenario. 
(2) Use determinants and Cramer’s rule to determine how many boys and how many girls were in the survey. 
(3) Answer the question in (2) by using inverse matrices. 
(4) Compare the two methods. When might one method be better than another method for solving a system? 
THESES ARE THE ANSWERS I RECIEVED 
A survey of 615 teenagers found that 44% of the boys and 35% of the girls would like to be taller. Altogether, 231 teenagers in the survey wished they were taller. 
(1) Set up a system of equations to model this scenario.
Let the number of boys be "b": 
Let the number of girls be "g":
EQUATION:
Quantity Equation.............:    b +     g = 615
"Want to be taller" Equation..:0.44b + 0.35g = 231 
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(2) Use determinants and Cramer’s rule to determine how many boys and how many girls were in the survey.
The determinant of the coefficients is (1*0.35)-(1*0.44) = -0.09
The "b" determinant is (615*0.35)-(1*231) = -15.7 
Then b = (b-determinant)/(coefficient determinant) = 175
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The "g" determinant is (1*231)-(615*0.44) = -39.6
Then g = (g-determinant)/(coefficient determinant) = 440
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b = 175
g = 440
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(3) Answer the question in (2) by using inverse matrices.
EQUATION:
Quantity Equation.............:    b +     g = 615
"Want to be taller" Equation..:0.44b + 0.35g = 231 
Write this pair of equations in matrix form:
..1......1....615
.0.44...0.35..231
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Find the inverse of the coefficient matrix:
inverse is
...-35/9.....100.9
....44.9....-100.9
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Multiply that inverse matrix times the column vector [615..231]
to get:
b = 175
g = 440
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(4) Compare the two methods. When might one method be better than another method for solving a system? 
The inverse matrix method is always faster if you have a calculator.
Cramer's method might be faster if you have only 2 equations with 2
variables and you don't make arithmetic mistakes.
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Cheers,
Stan H.