Chapter 8

Find the equation of the circle $$x^{2}+y^{2}+D x+E y+F=0$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. $$(0,0),(0,6),(3,3)$$

Use a system of equations to find the quadratic function \(f(x)=a x^{2}+b x+c\) that satisfies the equations. Solve the system using matrices. $$\begin{aligned} &f(-2)=-3\\\ &f(1)=-3\\\ &f(2)=-11 \end{aligned}$$

Factor the expression. $$4 y^{2}-28 y+49$$

The sums have been evaluated. Solve the given system for \(a\) and \(b\) to find the least squares regression line for the points. Use a graphing utility to confirm the results. $$\left\\{\begin{aligned} 6 b+15 a &=23.6 \\ 15 b+55 a &=48.8 \end{aligned}\right.$$

You are deciding how to invest a total of \(\$ 20,000\) in two funds paying \(5.5 \%\) and \(7.5 \%\) simple interest. You want to earn a total of \(\$ 1300\) in interest from the investments each year. (a) Write a system of equations in which one equation represents the total amount invested and the other equation represents the \(\$ 1300\) yearly interest. Let \(x\) and \(y\) represent the amounts invested at \(5.5 \%\) and \(7.5 \%,\) respectively. (b) Use a graphing utility to graph the two equations in the same viewing window. (c) How much of the \(\$ 20,000\) should you invest at \(5.5 \%\) to earn \(\$ 1300\) in interest per year? Explain your reasoning.

Find the equation of the circle $$x^{2}+y^{2}+D x+E y+F=0$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. $$(-3,-1),(2,4),(-6,8)$$

Use a system of equations to find the cubic function \(f(x)=a x^{3}+b x^{2}+c x+d\) that satisfies the equations. Solve the system using matrices. $$\begin{aligned} &f(-2)=-7\\\ &f(-1)=2\\\ &f(1)=-4\\\ &f(2)=-7 \end{aligned}$$

Solve the system of equations using the method of substitution or the method of elimination. $$\left\\{\begin{array}{c} 3 x-10 y=46 \\ x+y=-2 \end{array}\right.$$

The sums have been evaluated. Solve the given system for \(a\) and \(b\) to find the least squares regression line for the points. Use a graphing utility to confirm the results. $$\left\\{\begin{aligned} 7 b+21 a &=13.1 \\ 21 b+91 a &=-2.8 \end{aligned}\right.$$

You are offered two different rules for estimating the number of board feet in a 16 -foot log. (A board foot is a unit of measure for lumber equal to a board 1 foot square and 1 inch thick.) One rule is the Doyle Log Rule modeled by $$V=(D-4)^{2}, \quad 5 \leq D \leq 40$$ where \(D\) is the diameter (in inches) of the log and \(V\) is its volume (in board feet). The other rule is the Scribner Log Rule modeled by $$V=0.79 D^{2}-2 D-4, \quad 5 \leq D \leq 40$$ (a) Use a graphing utility to graph the two log rules in the same viewing window. (b) For what diameter do the two rules agree? (c) You are selling large logs by the board foot. Which rule would you use? Explain your reasoning.

Find the equation of the circle $$x^{2}+y^{2}+D x+E y+F=0$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. $$(0,0),(0,-2),(3,0)$$

Use a system of equations to find the cubic function \(f(x)=a x^{3}+b x^{2}+c x+d\) that satisfies the equations. Solve the system using matrices. $$\begin{aligned} &f(-2)=-17\\\ &f(-1)=-5\\\ &f(1)=1\\\ &f(2)=7 \end{aligned}$$

Solve the system of equations using the method of substitution or the method of elimination. $$\left\\{\begin{aligned} 5 x+7 y &=23 \\ -4 x-2 y &=-4 \end{aligned}\right.$$

Four test plots were used to explore the relationship between wheat yield \(y\) (in bushels per acre) and amount of fertilizer applied \(x\) (in hundreds of pounds per acre). The results are given by the ordered pairs \((1.0,32),(1.5,41),(2.0,48),\) and (2.5,53) (a) Find the least squares regression line \(y=a x+b\) for the data by solving the system for \(a\) and \(b\) \(\left\\{\begin{array}{l}4 b+7.0 a=174 \\ 7 b+13.5 a=322\end{array}\right.\) (b) Use the regression feature of a graphing utility to confirm the result in part (a). (c) Use the graphing utility to plot the data and graph the linear model from part (a) in the same viewing window. (d) Use the linear model from part (a) to predict the yield for a fertilizer application of 160 pounds per acre.

Algebraic-Graphical-Numerical The populations (in thousands) of Colorado \(C\) and Minnesota \(M\) from 2008 through 2012 can be modeled by the system $$\begin{aligned} &C=74.0 t+4303 \quad \text { Colorado }\\\ &M=33.0 t+4988 \quad \text { Minnesota } \end{aligned}$$ where \(t\) is the year, with \(t=8\) corresponding to 2008 (a) Record in a table the populations predicted by the models for the two states in the years 2013 through 2020 (b) According to the table, in what year(s) will the population of Colorado be greater than that of Minnesota? (c) Use a graphing utility to graph the models in the same viewing window. Estimate the point of intersection of the models. (d) Find the point of intersection algebraically. (e) Summarize your findings of parts (b) through (d).

A college student borrowed \(\$ 30,000\) to pay for tuition, room, and board. Some of the money was borrowed at \(4 \%,\) some at \(6 \%,\) and some at \(8 \% .\) How much was borrowed at each rate, given that the annual interest was \(\$ 1550\) and the amount borrowed at \(8 \%\) was three times the amount borrowed at \(6 \% ?\)

The currents in an electrical network are given by the solution of the system. $$\left\\{\begin{aligned} I_{1}-I_{2}+I_{3} &=0 \\ 2 I_{1}+2 I_{2} &=7 \\ 2 I_{2}+4 I_{3} &=8 \end{aligned}\right.$$ where \(I_{1}, I_{2},\) and \(I_{3}\) are measured in amperes. Solve the system of equations using matrices.

A small corporation borrowed \(\$ 775,000\) to expand its software line. Some of the money was borrowed at \(8 \%,\) some at \(9 \%,\) and some at \(10 \% .\) How much was borrowed at each rate, given that the annual interest was \(\$ 67,500\) and the amount borrowed at \(8 \%\) was four times the amount borrowed at \(10 \% ?\)

Finance A corporation borrowed \(1,500,000\) to expand its line of shoes. Some of the money was borrowed at \(3 \%,\) some at \(4 \%,\) and some at \(6 \% .\) Use a system of equations to determine how much was borrowed at each rate, given that the annual interest was \(74,000\) and the amount borrowed at \(6 \%\) was four times the amount borrowed at \(3 \% .\) Solve the system using matrices.

Determine whether the statement is true or false. Justify your answer. If a system of linear equations has two distinct solutions, then it has an infinite number of solutions.

Using Matrices A food server examines the amount of money earned in tips after working an 8 -hour shift. The server has a total of \(95\) in denominations of \( 1\) \( 5, 10,\) and \( 20\) bills. The total number of paper bills is 26\. The number of \( 5\) bills is 4 times the number of \(10\) bills, and the number of SI bills is I less than twice the number of \( 5\) bills. Write a system of linear equations to represent the situation. Then use matrices to find the number of each denomination.

Determine whether the statement is true or false. Justify your answer. If a system of linear equations has no solution, then the lines must be parallel.

Marketing A wholesale paper company sells a 100 -pound package of computer paper that consists of three grades, glossy, semi-gloss, and matte, for printing photographs. Glossy costs \( 5.50\) per pound, semigloss costs \(4.25\) per pound, and matte costs \(3.75\) per pound. One half of the 100 -pound package consists of the two less expensive grades. The cost of the 100 -pound package is \(480 .\) Set up and solve a system of equations, using matrices, to find the number of pounds of each grade of paper in a 100 -pound package.

Determine whether the statement is true or false. Justify your answer. Solving a system of equations graphically using a graphing utility always yields an exact solution.

When solving a system of equations by substitution, how do you recognize that the system has no solution?

True or False? Determine whether the statement is true or false. Justify your answer. Two matrices can be added only when they have the same dimension.

In the 2013 Women's NCAA Championship basketball game, the University of Connecticut defeated Louisville University by a score of 93 to \(60 .\) Connecticut won by scoring a combination of two-point baskets, three-point baskets, and one-point free throws. The number of two-point baskets was 12 more than the number of free throws. The number of free throws was three less than the number of threepoint baskets. What combination of scoring accounted for Connecticut's 93 points? (Source: NCAA)

Partial Fractions Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices. $$\frac{8 x^{2}}{(x-1)^{2}(x+1)}=\frac{A}{x+1}+\frac{B}{x-1}+\frac{C}{(x-1)^{2}}$$

Determine whether the statement is true or false. Justify your answer. Writing Briefly explain whether or not it is possible for a consistent system of linear equations to have exactly two solutions.

Find equations of lines whose graphs intersect the graph of the parabola \(y=x^{2}\) at (a) two points, (b) one point, and (c) no points. (There are many correct answers.)

The Augusta National Golf Club in Augusta, Georgia, is an 18 -hole course that consists of par- 3 holes, par-4 holes, and par-5 holes. A golfer who shoots par has a total of 72 strokes for the entire course. There are two more par- 4 holes than twice the number of par-5 holes, and the number of par-3 holes is equal to the number of par-5 holes. Find the numbers of par-3, par-4, and par-5 holes on the course. (Source: Augusta National, Inc.)

A video of the path of a ball thrown by a baseball player was analyzed with a grid covering the TV screen. The video was paused three times, and the position of the ball was measured each time. The coordinates obtained are shown in the table \((x \text { and } y\) are measured in feet). $$\begin{array}{|c|c|} \hline \text { Horizontal distance, } x & \text { Height, } y \\ \hline 0 & 5.0 \\ \hline 15 & 9.6 \\ \hline 30 & 12.4 \\ \hline \end{array}$$ (a) Use a system of equations to find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the points. Solve the system using matrices. (b) Use a graphing utility to graph the parabola. (c) Graphically approximate the maximum height of the ball and the point at which the ball strikes the ground. (d) Algebraically approximate the maximum height of the ball and the point at which the ball strikes the ground.

Think About It Find all value(s) of \(k\) for which the system of linear equations \(\left\\{\begin{array}{c}x+3 y=9 \\ 2 x+6 y=k\end{array}\right.\) has (a) infinitely many solutions and (b) no solution.

Create systems of two linear equations in two variables that have (a) no solution, (b) one distinct solution, and (c) infinitely many solutions.

When Kirchhoff's Laws are applied to the electrical network in the figure, the currents \(I_{1}, I_{2},\) and \(I_{3}\) (in amperes) are the solution of the system \(\left\\{\begin{aligned} I_{1}-I_{2}+I_{3} &=0 \\ 3 I_{1}+2 I_{2} &=7 \\\ 2 I_{2}+4 I_{3} &=8 \end{aligned}\right.\) Find the currents..

The table shows the average annual consumer costs \(y\) (in dollars) for health insurance from 2010 to 2012 . $$\begin{array}{|c|c|} \hline \text { Year } & \text { cost, } y \\ \hline 2010 & 1831 \\ \hline 2011 & 1922 \\ \hline 2012 & 2061 \\ \hline \end{array}$$ (a) Use a system of equations to find the equation of the parabola \(y=a t^{2}+b t+c\) that passes through the points. Let \(t\) represent the year, with \(t=0\) corresponding to \(2010 .\) Solve the system using matrices. (b) Use a graphing utility to graph the parabola and plot the data points. (c) Use the equation in part (a) to estimate the average consumer costs in \(2015,2020,\) and 2025 (d) Are your estimates in part (c) reasonable? Explain.

Create a system of linear equations in two variables that has the solution (2,-1) as its only solution. (There are many correct answers.)

A system of pulleys is loaded with 128 -pound and 32 -pound weights (see figure). The tensions \(t_{1}\) and \(t_{2}\) in the ropes and the acceleration \(a\) of the 32 -pound weight are modeled by the following system, where \(t_{1}\) and \(t_{2}\) are measured in pounds and \(a\) is in feet per second squared. Solve the system. $$\left\\{\begin{aligned} t_{1}-2 t_{2} &=0 \\ t_{1} &-2 a=128 \\ t_{2}+& a=32 \end{aligned}\right.$$

The table shows the average annual salaries \(y\) (in thousands of dollars) for public school classroom teachers in the United States from 2011 through 2013. $$\begin{array}{|c|c|} \hline \text { Year } & \text { Annual salary, } y \\ \hline 2011 & 55.5 \\ \hline 2012 & 55.4 \\ \hline 2013 & 56.4 \\ \hline \end{array}$$ (a) Use a system of equations to find the equation of the parabola \(y=a t^{2}+b t+c\) that passes through the points. Let \(t\) represent the year, with \(t=1\) corresponding to \(2011 .\) Solve the system using matrices. (b) Use a graphing utility to graph the parabola and plot the data points. (c) Use the equation in part (a) to estimate the average annual salaries in \(2015,2020,\) and 2025 (d) Are your estimates in part (c) reasonable? Explain.

Consider the system of equations. $$\left\\{\begin{array}{l} y=b^{x} \\ y=x^{b} \end{array}\right.$$ (a) Use a graphing utility to graph the system of equations for \(b=2\) and \(b=4\) (b) For a fixed value of \(b > 1,\) make a conjecture about the number of points of intersection of the graphs in part (a).

Let matrices \(A,B,C,\) and \(D\) be of dimensions \(2 \times 3,2 \times 3,3 \times 2,\) and \(2 \times 2\) respectively. Determine whether the matrices are of proper dimension to perform the operation(s). If so, give the dimension of the answer. \(C D\)

Fitting a Parabola To find the least squares regression parabola \(y=a x^{2}+b x+c\) for a set of points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) you can solve the following system of linear equations for \(a, b,\) and \(c\) $$\left\\{\begin{aligned} n c+\left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a &=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) c+\left(\sum_{i=1}^{n} x_{i}^{2}\right) b+\left(\sum_{i=1}^{n} x_{i}^{3}\right) a &=\sum_{i=1}^{n} x_{i} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}^{2}\right) c+\left(\sum_{i=1}^{n} x_{i}^{3}\right) b+\left(\sum_{i=1}^{n} x_{i}^{4}\right) a &=\sum_{i=1}^{n} x_{i}^{2} y_{i} \end{aligned}\right.$$ The sums have been evaluated. Solve the given system for \(a, b,\) and \(c\) to find the least squares regression parabola for the points. Use a graphing utility to confirm the result. $$\left\\{\begin{aligned} 4 c & \quad+40 a=19 \\ 40 b & \quad=-12 \\ 40 c & \quad+544 a=160 \end{aligned}\right.$$

Solve the system of equations for \(u\) and \(v\). While solving for these variables, consider the transcendental functions as constants. (Systems of this type are found in a course in differential equations.) \(\left\\{\begin{array}{l}u \sin x+v \cos x=0 \\ u \cos x-v \sin x=\sec x\end{array}\right.\)

Let matrices \(A,B,C,\) and \(D\) be of dimensions \(2 \times 3,2 \times 3,3 \times 2,\) and \(2 \times 2\) respectively. Determine whether the matrices are of proper dimension to perform the operation(s). If so, give the dimension of the answer. \(B C\)

Fitting a Parabola To find the least squares regression parabola \(y=a x^{2}+b x+c\) for a set of points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) you can solve the following system of linear equations for \(a, b,\) and \(c\) $$\left\\{\begin{aligned} n c+\left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a &=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) c+\left(\sum_{i=1}^{n} x_{i}^{2}\right) b+\left(\sum_{i=1}^{n} x_{i}^{3}\right) a &=\sum_{i=1}^{n} x_{i} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}^{2}\right) c+\left(\sum_{i=1}^{n} x_{i}^{3}\right) b+\left(\sum_{i=1}^{n} x_{i}^{4}\right) a &=\sum_{i=1}^{n} x_{i}^{2} y_{i} \end{aligned}\right.$$ The sums have been evaluated. Solve the given system for \(a, b,\) and \(c\) to find the least squares regression parabola for the points. Use a graphing utility to confirm the result. $$\left\\{\begin{aligned} 5 c & \quad+10 a=8 \\ 10 b & \quad=12 \\ 10 c & \quad+34 a=22 \end{aligned}\right.$$

Solve the system of equations for \(u\) and \(v\). While solving for these variables, consider the transcendental functions as constants. (Systems of this type are found in a course in differential equations.) \(\left\\{\begin{aligned} u \cos 2 x+v \sin 2 x &=0 \\ u(-2 \sin 2 x)+v(2 \cos 2 x) &=\csc 2 x \end{aligned}\right.\)

Write an equation of the line passing through the two points. Use the slope- intercept form, if possible. If not possible, explain why. $$(3,4),(10,6)$$

Let matrices \(A,B,C,\) and \(D\) be of dimensions \(2 \times 3,2 \times 3,3 \times 2,\) and \(2 \times 2\) respectively. Determine whether the matrices are of proper dimension to perform the operation(s). If so, give the dimension of the answer. $$C A-D$$

Fitting a Parabola To find the least squares regression parabola \(y=a x^{2}+b x+c\) for a set of points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) you can solve the following system of linear equations for \(a, b,\) and \(c\) $$\left\\{\begin{aligned} n c+\left(\sum_{i=1}^{n} x_{i}\right) b+\left(\sum_{i=1}^{n} x_{i}^{2}\right) a &=\sum_{i=1}^{n} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}\right) c+\left(\sum_{i=1}^{n} x_{i}^{2}\right) b+\left(\sum_{i=1}^{n} x_{i}^{3}\right) a &=\sum_{i=1}^{n} x_{i} y_{i} \\ \left(\sum_{i=1}^{n} x_{i}^{2}\right) c+\left(\sum_{i=1}^{n} x_{i}^{3}\right) b+\left(\sum_{i=1}^{n} x_{i}^{4}\right) a &=\sum_{i=1}^{n} x_{i}^{2} y_{i} \end{aligned}\right.$$ The sums have been evaluated. Solve the given system for \(a, b,\) and \(c\) to find the least squares regression parabola for the points. Use a graphing utility to confirm the result. $$\left\\{\begin{array}{c} 4 c+9 b+29 a=20 \\ 9 c+29 b+99 a=70 \\ 29 c+99 b+353 a=254 \end{array}\right.$$

Determine whether the statement is true or false. Justify your answer. When using Gaussian elimination to solve a system of linear equations, you may conclude that the system is inconsistent before you complete the process of rewriting the augmented matrix in row-echelon form.