Explanation Given information: The total concentration of pollutants is 6.0 ppm The concentration of carbon monoxide is 10 times of sulfur dioxide. The concentration of sulfur dioxide is equal to summation of nitric oxide and nitrogen dioxide. The total concentration sulfur dioxide nitric oxide is 0.8 ppm. Calculation: Consider x , y , z , and w are represents the concentration of sulfur dioxide, nitric oxide, nitrogen dioxide, and carbon monoxide respectively. Write the general form of equations using given information. x + y + z + w = 6 10 x − w = 0 x − y − z = 0 x + y = 0.8 Write the given equations in general matrix form [ A ] . [ A ] = [ 1 1 1 1 10 0 0 − 1 1 − 1 − 1 0 1 1 0 0 ] Here, [ A ] is the matrix form of coefficients x , y , z , and w and [ X ] is the constant of equations. Find the determinant | A | of matrix A . | A | = | 1 1 1 1 10 0 0 − 1 1 − 1 − 1 0 1 1 0 0 | Expand matrix A by second row. | A | = − 10 | 1 1 1 − 1 − 1 0 1 0 0 | + 0 − 0 + ( − 1 ) | 1 1 1 1 − 1 − 1 1 1 0 | = − 10 [ 1 ( 0 + 1 ) − 0 + 0 ] − 1 [ 1 ( − 1 + 1 ) − 1 ( − 1 − 1 ) + 0 ] = − 10 − 1 ( 0 + 2 ) = − 12 Replace the coefficients of x (elements of column 1) with constant [ X ] to form matrix [ X A ] . [ X A ] = [ 6 1 1 1 0 0 0 − 1 0 − 1 − 1 0 0.8 1 0 0 ] Find the determinant | X A | of matrix X A . | X A | = | 6 1 1 1 0 0 0 − 1 0 − 1 − 1 0 0.8 1 0 0 | Expand matrix X A by second column. | X A | = − 0 + 0 − 0 + ( − 1 ) | 6 1 1 0 − 1 − 1 0.8 1 0 | = − 1 [ 6 ( 0 + 1 ) − 0 + 0.8 ( − 1 + 1 ) ] = − 1 ( 6 + 0 ) = − 6 Find the concentration of sulfur dioxide ( x ) using the equation: x = | X A | | A | Substitute − 12 for | X A | and − 6 for | A | . x = − 6 − 12 = 0.5 ppm Therefore, the concentration of sulfur dioxide is 0.5 ppm _ . Replace the coefficients of y (elements of column 2) with constant [ X ] to form matrix [ X B ] . [ X B ] = [ 1 6 1 1 10 0 0 − 1 1 0 − 1 0 1 0.8 0 0 ] Find the determinant | X B | of matrix X B . | X B | = | 1 6 1 1 10 0 0 − 1 1 0 − 1 0 1 0.8 0 0 | Expand matrix X B by second row. | X B | = − 10 | 6 1 1 0 − 1 0 0.8 0 0 | + 0 − 0 + ( − 1 ) | 1 6 1 1 0 − 1 1 0

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

11th Edition

ISBN: 9780137554843

Author: Allyn Washington, Richard Evans

Publisher: PEARSON+

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Chapter 16.6, Problem 37E

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Chapter 16.6, Problem 36E

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Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

Ch. 16.1 - For matrices A and B, find A + B. Ch. 16.1 - Prob. 2PE Ch. 16.1 - Prob. 3PE Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E Ch. 16.1 - In Exercises 3–10, determine the value of the...Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E

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