Chapter 6

Describe one application of matrices.

Technology Use technology to find the solution. A pproximate values to the nearest thousandth. $$ \begin{aligned} 103 x-886 y+431 z &=1200 \\ -55 x+981 y &=1108 \\ -327 x+421 y+337 z &=99 \end{aligned} $$

Use elimination to solve the system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically or numerically. $$ \begin{array}{r} 2 x+y=15 \\ x-y=0 \end{array} $$

Use elimination to solve the system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically or numerically. $$ \begin{array}{l} x+3 y=10 \\ x-2 y=-5 \end{array} $$

Estimating the Weight of a Bear The following table shows the weight \(W\), neck size \(N\), and chest size \(C\) for a representative sample of black bears. $$ \begin{array}{ccc} \hline W \text { (pounds) } & N \text { (inches) } & C \text { (inches) } \\ \hline 100 & 17 & 27 \\ 272 & 25 & 36 \\ \hline 381 & 30 & 43 \end{array} $$ (a) Find values for \(a, b,\) and \(c\) so that the equation \(W=a+b N+c C\) models these data. (b) Estimate the weight of a bear with a 20 -inch neck and a 31 -inch chest size. (c) Explain why it is reasonable for the coefficients \(b\) and \(c\) to be positive.

Use elimination to solve the system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically or numerically. $$ \begin{array}{r} 4 x+2 y=10 \\ -2 x-y=10 \end{array} $$

Translations (Refer to the discussion in this section about translating a point.) Find a \(3 \times 3\) matrix A that performs the following translation of a point \((x, y)\) represented by \(X .\) Find \(A^{-1}\) and describe what it computes. Leontief Economic Model Suppose that a closed eco nomic region has three industries: service, electrical power and tourism. The service industry uses \(20 \%\) of its own production, \(40 \%\) of the electrical power, and \(80 \%\) of the tourism. The power company use \(40 \%\) of the service indus try, \(20 \%\) of the electrical power, and \(10 \%\) of the tourism The tourism industry uses \(40 \%\) of the service industry \(10 \%\) of the tourism. (A) Let \(S, E,\) and \(T\) be the numbers of units produced by the service, electrical, and tourism industries, respectively. The following system of linear equations can be used to determine the relative number of units each industry needs to produce. (This model assumes that all production is consumed by the region.) $$ \begin{aligned} &0.25+0.4 E+0.8 T=S\\\ &\begin{array}{l} 0.45+0.2 E+0.1 T=E \\ 0.45+0.4 E+0.1 T=T \end{array} \end{aligned} $$ Solve the system and write the solution in terms of \(T\). (B) If tourism produces 60 units, how many units should the service and electrical industries produce?

Three pumps are being used to empty a small swimming pool. The first pump is twice as fast as the second pump. The first two pumps can empty the pool in 8 hours, while all three pumps can empty it in 6 hours. How long would it take each pump to empty the pool individually? (Hint: Let \(x\) represent the fraction of the pool that the first pump can empty in 1 hour. Let \(y\) and \(z\) represent this fraction for the second and third pumps, respectively.)

Use elimination to solve the system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically or numerically. $$ \begin{array}{rr} x+y= & 500 \\ -x-y= & -500 \end{array} $$

Use elimination to solve the system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically or numerically. $$ \begin{aligned} &2 x+3 y=5\\\ &5 x-2 y=3 \end{aligned} $$

Discuss how to solve the matrix equation \(A X=B\) if \(A^{-1}\) exists

Use elimination to solve the system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically or numerically. $$ \begin{array}{l} 2 x+4 y=7 \\ -x-2 y=5 \end{array} $$

Give an example of a \(2 \times 2\) matrix \(A\) with only nonzero elements that does not have an inverse. Explain what happens if one attempts to find \(A^{-1}\) symbolically.

Use elimination to solve the system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically or numerically. $$ \begin{aligned} &4 x-3 y=5\\\ &3 x+4 y=2 \end{aligned} $$

Investment \(A\) sum of \(\$ 5000\) is invested in three mutual funds that pay \(8 \%, 11 \%,\) and \(14 \%\) annual interest rates. The amount of money invested in the fund paying \(14 \%\) equals the total amount of money invested in the other two funds, and the total annual interest from all three funds is \(\$ 595\) (a) Write a system of equations whose solution gives the amount invested in each mutual fund. Be sure to state what cach variable represents. (b) Solve the system of equations.

Use elimination to solve the system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically or numerically. $$ \begin{array}{r} 2 x+3 y=2 \\ x-2 y=-5 \end{array} $$

\(A\) sum of \(\$ 10,000\) is invested in three accounts that pay \(6 \%, 8 \%,\) and \(10 \%\) interest. Twice as much money is invested in the account paying \(10 \%\) as in the account paying \(6 \%,\) and the total annual interest from all three accounts is \(\$ 842\). (a) Write a system of equations whose solution gives the amount invested in each account. Be sure to state what each variable represents. (b) Solve the system of equations.

Use elimination to solve the system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically or numerically. $$ \begin{array}{c} x-3 y=1 \\ 2 x-6 y=2 \end{array} $$

Solve the system, if possible. $$ \begin{array}{r} \frac{1}{2} x-y=5 \\ x-\frac{1}{2} y=4 \end{array} $$

Solve the system, if possible. $$ \begin{aligned} &\frac{1}{2} x-\frac{1}{3} y=1\\\ &\frac{1}{3} x-\frac{1}{2} y=1 \end{aligned} $$

Each set of data can be modeled by \(f(x)=a x^{2}+b x+c\) (a) Write a linear system whose solution represents values of \(a, b,\) and \(c\) (b) Use technology to find the solution. (c) Graph \(f\) and the data in the same viewing rectangle. (d) Make your own prediction using \(f\). (Refer to the introduction to this section.) The table lists total iPod sales \(y\) in millions \(x\) years after 2004 $$ \begin{array}{cccc} x & 0 & 2 & 4 \\ y & 3 & 55 & 150 \end{array} $$

Solve the system, if possible. $$ \begin{array}{rr} 7 x-3 y= & -17 \\ -21 x+9 y= & 51 \end{array} $$

Each set of data can be modeled by \(f(x)=a x^{2}+b x+c\) (a) Write a linear system whose solution represents values of \(a, b,\) and \(c\) (b) Use technology to find the solution. (c) Graph \(f\) and the data in the same viewing rectangle. (d) Make your own prediction using \(f\). The table lists annual enrollment in thousands for the Head Start program \(x\) years after 1980 $$ \begin{array}{cccc} x & 0 & 10 & 26 \\ y & 376 & 541 & 909 \end{array} $$

Solve the system, if possible. $$ \begin{array}{rr} -\frac{1}{3} x+\frac{1}{6} y= & -1 \\ 2 x-y= & 6 \end{array} $$

Solve the system, if possible. $$ \begin{aligned} &\frac{2}{3} x+\frac{4}{3} y=\frac{1}{3}\\\ &-2 x-4 y=5 \end{aligned} $$

Each set of data can be modeled by \(f(x)=a x^{2}+b x+c\) (a) Write a linear system whose solution represents values of \(a, b,\) and \(c\) (b) Use technology to find the solution. (c) Graph \(f\) and the data in the same viewing rectangle. (d) Make your own prediction using \(f\). Carbon Dioxide Levels Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) is a greenhouse gas. The table at the top of the next page lists its concentration \(y\) in parts per million (ppm) measured at Mauna Loa, Hawaii, for three selected years \(x\). $$ \begin{array}{cccc} x & 1958 & 1973 & 2003 \\ y & 315 & 325 & 376 \end{array} $$

Solve the system, if possible. $$ \begin{array}{r} 5 x-2 y=7 \\ 10 x-4 y=6 \end{array} $$

A linear equation in three variables can be represented by a flat plane. Describe geometrically situations that can occur when a system of three linear equations has either no solution or an infinite number of solutions.

Solve the system, if possible. $$ \begin{array}{r} 0.2 x+0.3 y=8 \\ -0.4 x+0.2 y=0 \end{array} $$

Give an example of an augmented matrix in row-echelon form that represents a system of linear equations that has no solution. Explain your reasoning.

Solve the system, if possible. $$ \begin{array}{l} 2 x-3 y=1 \\ 3 x-2 y=2 \end{array} $$

Solve the system, if possible. $$ \begin{array}{rr} 2 x+3 y= & 7 \\ -3 x+2 y= & -4 \end{array} $$

Solve the system, if possible. $$ \begin{aligned} &5 x+4 y=-3\\\ &3 x-6 y=-6 \end{aligned} $$

Solve the system, if possible. $$ \begin{aligned} 7 x-5 y &=-15 \\ -2 x+3 y &=-2 \end{aligned} $$

Solve the system, if possible. $$ \begin{array}{rr} -5 x+3 y= & -36 \\ 4 x-5 y= & 34 \end{array} $$

Use elimination to solve the nonlinear system of equations. $$ \begin{aligned} &x^{2}+y=12\\\ &x^{2}-y=6 \end{aligned} $$

Use elimination to solve the nonlinear system of equations. $$ \begin{array}{r} x^{2}+2 y=15 \\ 2 x^{2}-y=10 \end{array} $$

Use elimination to solve the nonlinear system of equations. $$ \begin{aligned} &x^{2}+y^{2}=25\\\ &x^{2}+7 y=37 \end{aligned} $$

Use elimination to solve the nonlinear system of equations. $$ \begin{aligned} &x^{2}+y^{2}=36\\\ &x^{2}-6 y=36 \end{aligned} $$

Use elimination to solve the nonlinear system of equations. $$ \begin{array}{r} x^{2}+y^{2}=4 \\ 2 x^{2}+y^{2}=8 \end{array} $$

Use elimination to solve the nonlinear system of equations. $$ \begin{aligned} &x^{2}+y^{2}=4\\\ &x^{2}-y^{2}=4 \end{aligned} $$

Solve the nonlinear system of equations $$(a) symbolically\quad and\quad (b)\quad graphically.$$ $$ \begin{aligned} &x^{2}+y^{2}=16\\\ &x-y=0 \end{aligned} $$

Solve the nonlinear system of equations $$(a) symbolically\quad and\quad (b)\quad graphically.$$ $$ \begin{aligned} x^{2}-y &=1 \\ 3 x+y &=-1 \end{aligned} $$

Solve the nonlinear system of equations $$(a) symbolically\quad and\quad (b)\quad graphically.$$ $$ \begin{array}{r} x y=12 \\ x-y=4 \end{array} $$

Solve the nonlinear system of equations $$(a) symbolically\quad and\quad (b)\quad graphically.$$ $$ \begin{aligned} &x^{2}+y^{2}=2\\\ &x^{2}-y=0 \end{aligned} $$

Solve the system of linear equations $$(a) graphically,\quad (b) numerically,\quad and\quad (c) symbolically.$$ $$ \begin{array}{r} 2 x+y=1 \\ x-2 y=3 \end{array} $$

Solve the system of linear equations $$(a) graphically,\quad (b) numerically,\quad and\quad (c) symbolically.$$ $$ \begin{array}{l} 3 x+2 y=-2 \\ 2 x-y=-6 \end{array} $$

Solve the system of linear equations $$(a) graphically,\quad(b) numerically,\quad and\quad(c) symbolically.$$ $$\begin{array}{r}-2 x+y=0 \\\7 x-2 y=3\end{array}$$

Solve the system of linear equations $$(a) graphically,\quad(b) numerically,\quad and\quad(c) symbolically.$$ $$\begin{aligned}x-4 y &=15 \\\3 x-2 y &=15\end{aligned}$$

Approximate, to the nearest thousandth. any solutions to the nonlinear system of equations graphically. $$\begin{aligned}&x^{2}+y=5\\\&x+y^{2}=6\end{aligned}$$