Chapter 3

Fill in the blanks. The symbol \(\\{\) is called a left ____. It is used when writing a system of equations.

Determine whether the ordered triple is a solution of the system. $$ \begin{aligned} &(2,1,1)\\\ &\left\\{\begin{array}{l} x-y+z=2 \\ 2 x+y-z=4 \\ 2 x-3 y+z=2 \end{array}\right. \end{aligned} $$

A wood sculptor carves three types of statues with a chainsaw. The number of hours required for carving, sanding, and painting a totem pole, a bear, and a deer are shown in the table. How many of each should be produced to use all available labor hours? $$ \begin{array}{|l|c|c|c|c|} \hline & \text { Totem pole } & \text { Bear } & \text { Deer } & \text { Time available } \\ \hline \text { Carving } & 2 \mathrm{hr} & 2 \mathrm{hr} & 1 \mathrm{hr} & 14 \mathrm{hr} \\ \text { Sanding } & 1 \mathrm{hr} & 2 \mathrm{hr} & 2 \mathrm{hr} & 15 \mathrm{hr} \\ \text { Painting } & 3 \mathrm{hr} & 2 \mathrm{hr} & 2 \mathrm{hr} & 21 \mathrm{hr} \\ \hline \end{array} $$

Fill in the blanks. Fill in the blank. If the denominator determinant \(D\) for a system of equations is \(0,\) the equations of the system are dependent or the system is ___.

Disinfectants. A \(1 \%\) bleach solution is to be mixed with a \(5 \%\) bleach solution to obtain 15 ounces of a \(3 \%\) bleach solution. (TABLE CANT COPY)

Fill in the blanks: We read \(\\{(x, y) | x-5 y=9\\}\) as "the set of all \(\quad\) pairs \((x, y), \quad x-5 y=9 . "\)

Fill in the blanks. We read the set-builder notation \(\\{(x, y) | 3 x-5 y=1\\}\) as "the ____ of all ordered pairs \((x, y)\) ____ that \(3 x-5 y=1\).

Determine whether the ordered triple is a solution of the system. $$ \begin{aligned} &(-3,2,-1)\\\ &\left\\{\begin{array}{l} 3 x+y-z=-6 \\ 2 x+2 y+3 z=-1 \\ x+y+2 z=1 \end{array}\right. \end{aligned} $$

Jerry Rice, who played the majority of his career with the San Francisco 49 ers and the Oakland Raiders, holds the all-time record for touchdown (TD) passes caught. Here are some interesting facts about this feat. -He caught 30 more TD passes from Steve Young than he did from Joe Montana. -He caught 39 more TD passes from Joe Montana than he did from Rich Gannon. -He caught a total of 156 TD passes from Young, Montana, and Gannon. Determine the number of touchdown passes Rice has caught from Young, from Montana, and from Gannon.

Complete the evaluation of each determinant. $$ \begin{aligned} \left|\begin{array}{rr} 5 & -2 \\ -2 & 6 \end{array}\right| &=5(\quad-(-2)(-2)\\\ &=-4 \\ &=26 \end{aligned} $$

Represent each system using an augmented matrix. $$ \left\\{\begin{array}{l} x+2 y=6 \\ 3 x-y=-10 \end{array}\right. $$

Solve each system by substitution. See Examples 1 and 2 . $$ \left\\{\begin{array}{l} y=3 x \\ x+y=8 \end{array}\right. $$

Determine whether the ordered pair is a solution of the system of equations. See Example 1. $$ (-4,3) ;\left\\{\begin{array}{l} 4 x-y=-19 \\ 3 x+2 y=-6 \end{array}\right. $$

Determine whether the ordered triple is a solution of the system. $$ \begin{aligned} &(6,-7,-5)\\\ &\left\\{\begin{array}{l} 3 x-2 y-z=37 \\ x-3 y=27 \\ 2 x+7 y+2 z=-48 \end{array}\right. \end{aligned} $$

In 10 minutes, the top three finishers in the 2010 Nathan's Hot Dog Eating Contest consumed a total of 136 hot dogs. The winner, Joey Chestnut, ate 9 more hot dogs than the runner-up, Tim Janus. Pat Bertoletti finished a distant third, 8 hot dogs behind Janus. How many hot dogs did each person eat?

Complete the evaluation of each determinant. $$ \begin{aligned} \left|\begin{array}{ccc} 2 & 1 & 3 \\ 3 & 4 & 2 \\ 1 & 5 & 3 \end{array}\right| &=2\left|\begin{array}{cc} 4 & 3 \\ 5 & 3 \end{array}\right| &=2\left(\begin{array}{c} 1 \\ 1 \end{array}-10\right)-1(9-3)+3(15-5) \\ &=2(2)-1(0)+(11) \\ &=4-7+\\\ &=30 \end{aligned} $$

Write a system of two equations in two variables to solve each problem. Desserts. A slice of Mrs. Smith's apple pie and one scoop of Háagen-Dazs vanilla bean ice cream totals 600 calories. The pie has 20 more calories than the ice cream. Find the number of calories in each.

Represent each system using an augmented matrix. $$ \left\\{\begin{array}{l} x+y+z=4 \\ 2 x+y-z=1 \\ 2 x-3 y=1 \end{array}\right. $$

Solve each system by substitution. See Examples 1 and 2 . $$ \left\\{\begin{array}{l} y=x+2 \\ x+2 y=16 \end{array}\right. $$

Determine whether the ordered pair is a solution of the system of equations. See Example 1. $$ (-1,2) ;\left\\{\begin{array}{l} 3 x-y=-5 \\ x-y=-4 \end{array}\right. $$

Determine whether the ordered triple is a solution of the system. $$ \begin{aligned} &(-4,0,9)\\\ &\left\\{\begin{array}{l} x+2 y-3 z=-31 \\ 2 x+6 z=46 \\ 3 x-y=-12 \end{array}\right. \end{aligned} $$

Evaluate each determinant. $$ \left|\begin{array}{ll} 2 & 3 \\ 2 & 5 \end{array}\right| $$

Write a system of two equations in two variables to solve each problem. Area Codes. The entire state of Montana has just one telephone area code. The same is true for Idaho. The sum of their area codes is 614 and the difference is \(198,\) and Montana has the numerically larger one. Find the area code of each of these states.

For each augmented matrix, give the system of equations that it represents. $$ \left[\begin{array}{lll} 1 & 6 & 7 \\ 0 & 1 & 4 \end{array}\right] $$

Solve each system by substitution. See Examples 1 and 2 . $$ \left\\{\begin{array}{l} x=2+y \\ 2 x+y=13 \end{array}\right. $$

Determine whether the ordered pair is a solution of the system of equations. See Example 1. $$ (2,-3) ;\left\\{\begin{array}{l} y+2=\frac{1}{2} x \\ 3 x+2 y=0 \end{array}\right. $$

Solve each system. $$ \left\\{\begin{array}{l} x+y+z=4 \\ 2 x+y-z=1 \\ 2 x-3 y+z=1 \end{array}\right. $$

Evaluate each determinant. $$ \left|\begin{array}{ll} 3 & 2 \\ 2 & 4 \end{array}\right| $$

Write a system of two equations in two variables to solve each problem. Hiking. The Pacific Crest Trail runs from the U.S. border with Mexico to its border with Canada. The Appalachian Trail extends between Springer Mountain in Georgia and Mount Katahdin in Maine. The sum of the lengths of the trails is \(4,824\) miles, the difference is 476 miles, and the Pacific Crest Trail is the longer. Find the length of each trail. (IMAGE CANT COPY)

For each augmented matrix, give the system of equations that it represents. $$ \left[\begin{array}{rrrr} 1 & -2 & 9 & 1 \\ 0 & 1 & 4 & 0 \\ 0 & 0 & 1 & -7 \end{array}\right] $$

Solve each system by substitution. See Examples 1 and 2 . $$ \left\\{\begin{array}{l} x=-5+y \\ 3 x-2 y=-7 \end{array}\right. $$

Determine whether the ordered pair is a solution of the system of equations. See Example 1. $$ (1,2) ;\left\\{\begin{array}{l} 2 x-y=0 \\ y=\frac{1}{2} x+\frac{3}{2} \end{array}\right. $$

Solve each system. $$ \left\\{\begin{array}{l} x+y+z=4 \\ x-y+z=2 \\ x-y-2 z=-1 \end{array}\right. $$

The sum of the measures of the angles of any triangle is \(180^{\circ} .\) In \(\Delta A B C, \angle A\) measures \(100^{\circ}\) less than the sum of the measures of \(\angle B\) and \(\angle C,\) and the measure of \(\angle C\) is \(40^{\circ}\) less than twice the measure of \(\angle B .\) Find the measure of each angle of the triangle.

Evaluate each determinant. $$ \left|\begin{array}{rr} -9 & 7 \\ 4 & -2 \end{array}\right| $$

In \(2009,\) there was a combined total of \(4,046\) Gap and Aéropostale clothing stores worldwide. The number of Gap stores was \(3 \frac{1}{4}\) times more than the number of Aéropostale stores. How many Gap stores and how many Aéropostale stores were there that year'? (Source: wikinvest.com)

Perform each of the following elementary row operations on the augumented matrix \(\left[\begin{array}{rrr}-3 & 1 & -6 \\ 1 & -4 & 4\end{array}\right].\) $$ R_{1} \leftrightarrow R_{2} $$

Solve each system by substitution. See Examples 1 and 2 . $$ \left\\{\begin{array}{l} x+2 y=6 \\ 3 x-y=-10 \end{array}\right. $$

Determine whether the ordered pair is a solution of the system of equations. See Example 1. $$ \left(\frac{1}{2}, \frac{1}{3}\right) ;\left\\{\begin{array}{l} 2 x+3 y=2 \\ 4 x-9 y=1 \end{array}\right. $$

Solve each system. $$ \left\\{\begin{array}{l} 3 x+2 y-5 z=3 \\ 4 x-2 y-3 z=-10 \\ 5 x-2 y-2 z=-11 \end{array}\right. $$

A quadrilateral is a four-sided polygon. The sum of the measures of the angles of any quadrilateral is \(360^{\circ}\). In the illustration, the measures of \(\angle A\) and \(\angle B\) are the same. The measure of \(\angle C\) is \(20^{\circ}\) greater than the measure of \(\angle A\) and the measure of \(\angle D\) is \(60^{\circ}\) less than \(\angle B .\) Find the measure of \(\angle A, \angle B, \angle C,\) and \(\angle D\)

Evaluate each determinant. $$ \left|\begin{array}{rr} -1 & 2 \\ 3 & -4 \end{array}\right| $$

Avalanches. For the \(2009-2010\) snow season, the total number of avalanche fatalities in the United States and Canada was 48 . If the number in the United States was three times greater than the number in Canada, how many avalanche fatalities were there in each country? (Source: Avalanche.org)

Determine whether the ordered pair is a solution of the system of equations. See Example 1. $$ \left(-\frac{3}{4}, \frac{2}{3}\right) ;\left\\{\begin{array}{l} 4 x+3 y=-1 \\ 4 x-3 y=-5 \end{array}\right. $$

Solve each system. $$ \left\\{\begin{array}{l} 5 x+4 y+2 z=-2 \\ 3 x+4 y-3 z=-27 \\ 2 x-4 y-7 z=-23 \end{array}\right. $$

\(X\) -Files, Will \& Grace, and Seinfeld are three of the most popular television shows of all time. The total number of episodes of these three shows is \(575 .\) There are 21 more episodes of \(X\) -Files than Seinfeld , and the difference between the number of episodes of Will \& Grace and Seinfeld is 14 Find the number of episodes of each show.

Evaluate each determinant. $$ \left|\begin{array}{rr} 5 & 20 \\ 10 & 6 \end{array}\right| $$

Perform each of the following elementary row operations on the augumented matrix \(\left[\begin{array}{rrr}-3 & 1 & -6 \\ 1 & -4 & 4\end{array}\right].\) $$ -\frac{1}{3} R_{1} $$

Solve each system by substitution. See Examples 1 and 2 . $$ \left\\{\begin{array}{l} 0.3 a+0.1 b=0.5 \\ \frac{4}{3} a+\frac{1}{3} b=3 \end{array}\right. $$

Determine whether the ordered pair is a solution of the system of equations. See Example 1. $$ (-0.2,0.5) ;\left\\{\begin{array}{l} 2 x+5 y=2.1 \\ 5 x+y=-0.5 \end{array}\right. $$