Chapter 6

If possible, solve the nonlinear system of equations. $$ \begin{aligned} &x^{2}-y=3\\\ &x+y=3 \end{aligned} $$

Complete the following for the given system of linear equations. (a) Write the system in the form \(A X=B\). (b) Solve the linear system by computing \(X=A^{-1} B\) with a calculator. Approximate the solution to the $$ \begin{array}{rr} 0.08 x-0.7 y= & -0.504 \\ 1.1 x-0.05 y= & 0.73 \end{array} $$

If possible, solve the nonlinear system of equations. $$ \begin{array}{r} x y=8 \\ x+y=6 \end{array} $$

Maximizing Revenue\(\quad\) A refinery produces both gasoline and fuel oil, and sells gasoline for \(\$ 4,00\) per gallon and fuel oil for \(\$ 3.60\) per gallon. The refinery can produce at most \(600,000\) gallons a day, but must produce at least 2 gallons of fuel oil for every gallon of gasoline. At least \(150,000\) gallons of fuel oil must be made each day for the coming winter. Determine how much of each type of fuel should be produced to maximize revenue.

Complete the following for the given system of linear equations. (a) Write the system in the form \(A X=B\). (b) Solve the linear system by computing \(X=A^{-1} B\) with a calculator. Approximate the solution to the $$ \begin{aligned} -231 x+178 y &=-439 \\ 525 x-329 y &=2282 \end{aligned} $$

If possible, solve the nonlinear system of equations. $$ \begin{array}{r} 2 x-y=0 \\ 2 x y=4 \end{array} $$

Minimizing cost (Refer to Example \(6 .\) ) A breeder is mixing Brand \(A\) and Brand \(B\). Each serving should contain at least 60 grams of protein and 30 grams of fat. Brand A costs 80 cents per unit, and Brand B costs 50 cents per unit. Each unit of Brand A contains 15 grams of protein and 10 grams of fat, whereas each unit of Brand B contains 20 grams of protein and 5 grams of fat. Determine how much of each food should be bought to achieve a minimum cost per serving.

Complete the following for the given system of linear equations. (a) Write the system in the form \(A X=B\). (b) Solve the linear system by computing \(X=A^{-1} B\) with a calculator. Approximate the solution to the $$ \begin{array}{rr} 3.1 x+1.9 y-z= & 1.99 \\ 6.3 x & -9.9 z=-3.78 \\ -x+1.5 y+7 z= & 5.3 \end{array} $$

Reduced Row-Rchelon Form Use Gauss-Jordan elimination to solve the system of equations. $$ \begin{aligned} &x-y=1\\\ &x+y=5 \end{aligned} $$

If possible, solve the nonlinear system of equations. $$ \begin{aligned} x^{2}+y^{2} &=20 \\ y &=2 x \end{aligned} $$

Pet Food Cost A pet owner is buying two brands of food, \(\mathbf{X}\) and \(\mathbf{Y},\) for his animals. Each serving of the mixture of the two foods should contain at least 60 grams of protein and 40 grams of fat. Brand X costs 75 cents per unit, and Brand Y costs 50 cents per unit. Each unit of Brand X contains 20 grams of protein and 10 grams of fat, whereas each unit of Brand Y contains 10 grams of protein and 10 grams of fat. How much of each brand should be bought to obtain a minimum cost per serving?

Complete the following for the given system of linear equations. (a) Write the system in the form \(A X=B\). (b) Solve the linear system by computing \(X=A^{-1} B\) with a calculator. Approximate the solution to the $$ \begin{aligned} 17 x-22 y-19 z &=-25.2 \\ 3 x+13 y-9 z &=105.9 \\ x-2 y+6.1 z &=-23.55 \end{aligned} $$

Reduced Row-Rchelon Form Use Gauss-Jordan elimination to solve the system of equations. $$ \begin{array}{rr} 2 x+3 y= & 1 \\ x-2 y= & -3 \end{array} $$

If possible, solve the nonlinear system of equations. $$ \begin{aligned} &x^{2}+y^{2}=9\\\ &x+y=3 \end{aligned} $$

Digititing Letters Complete the following. (a) Design a matrix A with dimension \(4 \times 4\) that represents a digital image of the given letter. Asswme that there are four gray levels from 0 to 3 . (b) Find a matrix \(B\) such that \(B-A\) represents the negative image of the picture represented by matrix \(\mathbf{A}\) from part ( \(a\) ). $$\mathbf{z}$$

Raising Animals \(\mathbf{A}\) breeder can raise no more than 50 hamsters and mice and no more than 20 hamsters. If she sells the hamsters for \(\$ 15\) each and the mice for \(\$ 10\) each, find the maximum revenue produced.

Complete the following for the given system of linear equations. (a) Write the system in the form \(A X=B\). (b) Solve the linear system by computing \(X=A^{-1} B\) with a calculator. Approximate the solution to the $$ \begin{array}{rr} 3 x-y+z= & 4.9 \\ 5.8 x-2.1 y & =-3.8 \\ -x & +2.9 z=3.8 \end{array} $$

Reduced Row-Rchelon Form Use Gauss-Jordan elimination to solve the system of equations. $$ \begin{array}{rr} x+2 y+z= & 3 \\ y-z= & -2 \\ -x-2 y+2 z= & 6 \end{array} $$

If possible, solve the nonlinear system of equations. $$ \begin{array}{r} \sqrt{x}-2 y=0 \\ x-y=-2 \end{array} $$

Digititing Letters Complete the following. (a) Design a matrix A with dimension \(4 \times 4\) that represents a digital image of the given letter. Asswme that there are four gray levels from 0 to 3 . (b) Find a matrix \(B\) such that \(B-A\) represents the negative image of the picture represented by matrix \(\mathbf{A}\) from part ( \(a\) ). $$\mathbf{N}$$

Maximizing Storage \(\mathbf{A}\) manager wants to buy filing cabinets. Cabinet \(\mathbf{X}\) costs \(\$ 100\), requires 6 square feet of floor space, and holds 8 cubic feet. Cabinet Y costs \(\$ 200,\) requires 8 square feet of floor space, and holds 12 cubic feet. No more than \(\$ 1400\) can be spent, and the office has room for no more than 72 square feet of cabinets. The manager wants the maximum storage capacity within the limits imposed by funds and space. How many of each type of cabinet should be bought?

Complete the following for the given system of linear equations. (a) Write the system in the form \(A X=B\). (b) Solve the linear system by computing \(X=A^{-1} B\) with a calculator. Approximate the solution to the $$ \begin{array}{rr} 1.2 x-0.3 y-0.7 z= & -0.5 \\ -0.4 x+1.3 y+0.4 z= & 0.9 \\ 1.7 x+0.6 y+1.1 z= & 1.3 \end{array} $$

Reduced Row-Rchelon Form Use Gauss-Jordan elimination to solve the system of equations. $$ \begin{array}{rr} x+z= & 2 \\ x-y-z= & 0 \\ -2 x+y & =-2 \end{array} $$

If possible, solve the nonlinear system of equations. $$ \begin{array}{rr} x^{2}+y^{2}= & 4 \\ 2 x^{2}+y & =-3 \end{array} $$

Maximizing Profit\(\quad\) A business manufactures two parts, \(\mathbf{X}\) and \(\mathbf{Y}\). Machines \(\mathbf{A}\) and \(\mathbf{B}\) are needed to make each part. To make part \(\mathbf{X},\) machine \(\mathbf{A}\) is needed for 4 hours and machine \(B\) is needed for 2 hours. To make part Y, machine A is needed for 1 hour and machine B is needed for 3 hours. Machine \(A\) is available for 40 hours each week and machine \(\mathbf{B}\) is available for 30 hours. The profit from part \(\mathbf{X}\) is \(\$ 500\) and the profit from part \(\mathbf{Y}\) is S600. How many parts of each type should be made to maximize weekly profit?

Translations (Refer to the discussion in this section about translating a point.) The matrix product AX performs a translation on the point \((x, y),\) where $$ A=\left[\begin{array}{lll} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{array}\right] \text { and } \quad X=\left[\begin{array}{l} x \\ y \\ 1 \end{array}\right] $$ (A) Predict the new location of the point (x, y) when it is translated by \(A\). Compute \(Y=A X\) to verify your prediction. (B) Make a conjecture as to what \(A^{-1}\) Y represents. Find \(A^{-1}\) and calculate \(A^{-1} Y\) to test your conjecture. (C) What will \(A A^{-1}\) and \(A^{-1} A\) equal? $$ A=\left[\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{array}\right],(x, y)=(0,1), \text { and } X=\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right] $$

Reduced Row-Rchelon Form Use Gauss-Jordan elimination to solve the system of equations. $$ \begin{array}{rr} x-y+2 z= & 7 \\ 2 x+y-4 z= & -27 \\ -x+y-z= & 0 \end{array} $$

If possible, solve the nonlinear system of equations. $$ \begin{array}{rr} 2 x^{2}-y= & 5 \\ -4 x^{2}+2 y= & -10 \end{array} $$

Minimizing Cost Two substances, \(\mathbf{X}\) and \(\mathbf{Y}\), are found in pet food. Each substance contains the ingrodients A and B. Substance \(X\) is \(20 \%\) ingredient \(A\) and \(50 \%\) ingredient B. Substance \(Y\) is \(50 \%\) ingredient \(A\) and \(30 \%\) ingredient \(\mathbf{B}\). The cost of substance \(\mathbf{X}\) is \(\$ 2\) per pound, and the cost of substance \(Y\) is \(\$ 3\) per pound. The pet store needs at least 251 pounds of ingredient \(A\) and at least 200 pounds of ingredient \(B\). If cost is to be minimal, how many pounds of each substance should be ordered? Find the minimum cost.

Translations (Refer to the discussion in this section about translating a point.) The matrix product AX performs a translation on the point \((x, y),\) where $$ A=\left[\begin{array}{lll} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{array}\right] \text { and } \quad X=\left[\begin{array}{l} x \\ y \\ 1 \end{array}\right] $$ (A) Predict the new location of the point (x, y) when it is translated by \(A\). Compute \(Y=A X\) to verify your prediction. (B) Make a conjecture as to what \(A^{-1}\) Y represents. Find \(A^{-1}\) and calculate \(A^{-1} Y\) to test your conjecture. (C) What will \(A A^{-1}\) and \(A^{-1} A\) equal? $$ A=\left[\begin{array}{rrr} 1 & 0 & -4 \\ 0 & 1 & 5 \\ 0 & 0 & 1 \end{array}\right],(x, y)=(4,2), \text { and } X=\left[\begin{array}{l} 4 \\ 2 \\ 1 \end{array}\right] $$

Reduced Row-Rchelon Form Use Gauss-Jordan elimination to solve the system of equations. $$ \begin{aligned} 2 x-4 y-6 z &=2 \\ x-3 y+z &=12 \\ 2 x+y+3 z &=5 \end{aligned} $$

If possible, solve the nonlinear system of equations. $$ \begin{array}{rr} -6 \sqrt{x}+2 y= & -3 \\ 2 \sqrt{x}-\frac{2}{3} y= & 1 \end{array} $$

Give the general form of a system of linear inequalities in two variables. Discuss what distinguishes a system of linear inequalities from a nonlinear system of inequalities.

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. 3 units to the left and 5 units downward

Reduced Row-Rchelon Form Use Gauss-Jordan elimination to solve the system of equations. $$ \begin{aligned} 2 x+y-z &=2 \\ x-2 y+z &=0 \\ x+3 y-2 z &=4 \end{aligned} $$

If possible, solve the nonlinear system of equations. $$ \begin{aligned} &x^{2}-y=4\\\ &x^{2}+y=4 \end{aligned} $$

Discuss how to use test points to solve a linear inequality. Give an example.

Reduced Row-Rchelon Form Use Gauss-Jordan elimination to solve the system of equations. $$ \begin{aligned} -2 x-y+z &=3 \\ x+y-3 z &=1 \\ x-2 y-4 z &=2 \end{aligned} $$

If possible, solve the nonlinear system of equations. $$ \begin{array}{r} x^{2}+x=y \\ 2 x^{2}-y=2 \end{array} $$

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. (Refer to Example 2.) The matrix \(B\) rotates the point \((x, y)\) clockwise about the origin \(45^{\circ},\) where $$ B=\left[\begin{array}{rrr} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{array}\right] \text { and } B^{-1}=\left[\begin{array}{rrr} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{array}\right] $$ (A) Let \(X\) represent the point \((-\sqrt{2},-\sqrt{2}) .\) Compute \(\boldsymbol{Y}=\boldsymbol{B} \boldsymbol{X}\) (B) Find \(B^{-1} Y\). Interpret what \(B^{-1}\) computes.

If possible, solve the nonlinear system of equations. $$ \begin{aligned} &x^{3}-x=3 y\\\ &x-y=0 \end{aligned} $$

If possible, solve the nonlinear system of equations. $$ \begin{aligned} x^{4}+y &=4 \\ 3 x^{2}-y &=0 \end{aligned} $$

The matrix \(A\) translates a point to the right 4 units and downward 2 units, and the matrix \(B\) translates a point to the left 3 units and upward 3 units, where $$ A=\left[\begin{array}{rrr} 1 & 0 & 4 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{array}\right] \text { and } B=\left[\begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{array}\right] $$ (A) Let \(X\) represent the point \((1,1)\). Predict the result of \(\boldsymbol{Y}=\boldsymbol{A B X} .\) Check your prediction. (B) Find \(A B\) mentally, and then compute \(A B\) (C) Would you expect \(A B=B A ?\) Verify your answer. (D) Find \((A B)^{-1}\) mentally. Explain your reasoning.

Technology Use technology to find the solution. A pproximate values to the nearest thousandth. $$ \begin{array}{l} 2.1 x+0.5 y+1.7 z=4.9 \\ -2 x+1.5 y-1.7 z=3.1 \\ 5.8 x-4.6 y+0.8 z=9.3 \end{array} $$

The area of a rectangle with length l and width \(w\) is computed by \(A(l, w)=l w,\) and its perimeter is calculated by \(P(l, w)=2 l+2 w .\) Assume that \(l>w\) and use the method of substitution to solve the system of equations for \(l\) and \(w\). $$ \begin{array}{l} A(l, w)=35 \\ P(l, w)=24 \end{array} $$

Technology Use technology to find the solution. A pproximate values to the nearest thousandth. $$ \begin{array}{l} 53 x+95 y+12 z=108 \\ 81 x-57 y-24 z=-92 \\ -9 x+11 y-78 z=21 \end{array} $$

The area of a rectangle with length l and width \(w\) is computed by \(A(l, w)=l w,\) and its perimeter is calculated by \(P(l, w)=2 l+2 w .\) Assume that \(l>w\) and use the method of substitution to solve the system of equations for \(l\) and \(w\). $$ \begin{aligned} &A(l, w)=300\\\ &P(l, w)=70 \end{aligned} $$

Discuss whether matrix multiplication is more like multiplication of functions or composition of functions. Explain your reasoning.

Technology Use technology to find the solution. A pproximate values to the nearest thousandth. $$ \begin{aligned} 0.1 x+0.3 y+1.7 z &=0.6 \\ 0.6 x+0.1 y-3.1 z &=6.2 \\ 2.4 y+0.9 z &=3.5 \end{aligned} $$

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. $$ \begin{aligned} &x+y=20\\\ &x-y=8 \end{aligned} $$