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Chapter 8: Problem 2
Two systems of equations that have the same solution set are called _____ systems.
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Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form . Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. $$\left[\begin{array}{rrrr} -4 & 1 & 0 & 6 \\ 1 & -2 & 3 & -4 \end{array}\right]$$
When solving a system of equations by substitution, how do you recognize that the system has no solution?
Sales The projected sales \(S\) (in millions of dollars) of two clothing retailers from 2015 through 2020 can be modeled by \(\left\\{\begin{array}{ll}S-149.9 t=415.5 & \text { Retailer } \mathrm{A} \\\ S-183.1 t=117.3 & \text { Retailer } \mathrm{B}\end{array}\right.\) where \(t\) is the year, with \(t=5\) corresponding to 2015 (a) Solve the system of equations using the method of your choice. Explain why you chose that method. (b) Interpret the meaning of the solution in the context of the problem. (c) Interpret the meaning of the coefficient of the \(t\) -term in each model. (d) Suppose the coefficients of \(t\) were equal and the models remained the same otherwise. How would this affect your answers in parts (a) and (b)?
Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution. $$\left\\{\begin{aligned} x+2 y &=7 \\ 2 x+y &=8 \end{aligned}\right.$$
Chemistry Thirty liters of a \(40 \%\) acid solution are obtained by mixing a \(25 \%\) solution with a \(50 \%\) solution. (a) Write a system of equations in which one equation represents the amount of final mixture required and the other represents the percent of acid in the final mixture. Let \(x\) and \(y\) represent the amounts of the \(25 \%\) and \(50 \%\) solutions, respectively. (b) Use a graphing utility to graph the two equations in part (a) in the same viewing window. (c) As the amount of the \(25 \%\) solution increases, how does the amount of the \(50 \%\) solution change? (d) How much of each solution is required to obtain the specified concentration of the final mixture?
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