Chapter 3

Fill in the blanks. If a point lies on the graph of an equation, it is a solution of the equation, and the coordinates of the point _____ the equation.

Fill in the blanks. A parallelogram is a four-sided figure with two pairs of ______ sides.

Fill in the blanks. \(\left|\begin{array}{rr}4 & 9 \\ -6 & 1\end{array}\right|\) is a ___ . The numbers 4 and 1 lie along its main ___.

Fill in the blanks. A ___ is a rectangular array of numbers written within brackets.

Fill in the blanks. \(A x+B y=C\) is the ________ form of a linear equation.

Fill in the blanks. \(\left\\{\begin{array}{l}2 x+y-3 z=0 \\ 3 x-y+4 z=5 \\ 4 x+2 y-6 z=0\end{array}\right.\) is called a _____ of three linear equations in three variables. Each equation is written in ____ \(A x+B y+C z=D\) form.

Fill in the blanks. \(\left\\{\begin{array}{l}x-2 y=4 \\ 2 x-y=3\end{array}\right.\) is called a _____ of linear equations.

Fill in the blanks. The process of determining an equation whose graph contains given points is called curve _____.

Fill in the blanks. Suppose a hammer can be manufactured in two different ways. The number of hammers that will cost equal amounts to produce either way is called the_____point.

Fill in the blanks. A determinant is number that is associated with a __ matrix.

Fill in the blanks. Each number in a matrix is called an ____ or entry of the matrix.

Fill in the blanks. When a system of equations has at least one solution, it is called a ____ system. If a system has no solutions, it is called an ____ system.

Fill in the blanks. In the equation \(x+3 y=-1,\) the \(x\) -term has an understood _____________ of 1

Write a system of three equations in three variables that models the situation. Do not solve the system. A bakery makes three kinds of pies: chocolate cream, which sells for \(\$ 5\); apple, which sells for \(\$ 6\); and cherry, which sells for \(\$ 7 .\) The cost to make the pies is \(\$ 2, \$ 3,\) and \(\$ 4\) respectively. Let \(x=\) the number of chocolate cream pies made daily, \(y=\) the number of apple pies made daily, and \(z=\) the number of cherry pies made daily. -Each day, the bakery makes 50 pies. -Each day, the revenue from the sale of the pies is \(\$ 295\). -Each day, the cost to make the pies is \(\$ 145 .\)

Fill in the blanks. $$ \text { The } \\__\text { of } b_{1} \text { in }\left|\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right| \text { is }\left|\begin{array}{ll} a_{2} & c_{2} \\ a_{3} & c_{3} \end{array}\right| $$

Fill in the blanks. If two equations have different graphs, they are called ____ equations. Two equations with the same graph are called ____ equations.

Fill in the blanks. When we add the two equations of the system \(\left\\{\begin{array}{l}x+y=5 \\\ x-y=-3\end{array}\right.\) the \(y\) -terms are _____________.

Fill in the blanks. Solutions of a system of three equations in three variables, \(x, y\) and \(z,\) are written in the form \((x, y, z)\) and are called ordered ____.

Write a system of three equations in three variables that models the situation. Do not solve the system. Let \(x=\) the number of calories in a Big Mac hamburger, \(y=\) the number of calories in a small order of French fries, and \(z=\) the number of calories in a medium Coca-Cola. -The total number of calories in a Big Mac hamburger, a small order of French fries, and a medium Coke is \(1,000\). -The number of calories in a Big Mac is 260 more than in a small order of French fries. -The number of calories in a small order of French fries is 40 more than in a medium Coke. (Source: McDonald's USA)

A company charges a \(\$ 75\) setup fee plus \(\$ 5.25\) per shirt to silkscreen a design on specialty t-shirts. Write an equation that gives the cost of purchasing \(x\) shirts.

Fill in the blanks. ___rule uses determinants to solve systems of linear equations.

Fill in the blanks. A matrix that represents the equations of a system is called an ______ matrix.

Fill in the blanks. When solving a system of two linear equations by the graphing method, we look for the point of ____ of the two lines.

To solve \(\left\\{\begin{array}{l}y=3 x+1 \\ x+y=4\end{array}\right.\) we can ____________ \(3 x+1\) for \(y\) in the second equation.

Fill in the blanks. The graph of the equation \(2 x+3 y+4 z=5\) is a flat surface called a _____.

What equation results when the coordinates of the point \((2,-3)\) are substituted into \(y=a x^{2}+b x+c ?\)

Fill in the blanks. $$ \left|\begin{array}{ll} a & b \\ c & d \end{array}\right|= __ $$

Fill in the blanks. Elementary ______ operations can be used on an augmented matrix to produce a simpler equivalent matrix that gives the solution of a system. This process is called ____ - _____ elimination.

If the system \(\left\\{\begin{array}{l}4 x-3 y=7 \\ 3 x-y=6\end{array}\right.\) is to be solved using the substitution method, what variable in what equation would it be easier to solve for?

The equation \(y=5 x^{2}-6 x+1\) is written in the form \(y=a x^{2}+b x+c .\) What are \(a, b,\) and \(c ?\)

Fill in the blanks. To find the minor of \(5,\) we cross out the elements of the determinant that are in the same row and column as ___ $$ \left|\begin{array}{rrr} 3 & 5 & 1 \\ 6 & -2 & 2 \\ 8 & -1 & 4 \end{array}\right| $$

a. Write an expression that represents the total value of \(x\) ounces of ginseng tea that costs \(\$ 32\) per pound. b. Write an expression that represents the amount of hydrochloric acid in \(x\) gallons of a \(3 \%\) hydrochloric acid solution.

Given the equation \(3 x+y=-4\) a. solve for \(x\) b. solve for \(y\) c. Which variable was easier to solve for? Explain why.

Fill in the blanks. In evaluating the determinant below, about what row or column was it expanded? $$ \left|\begin{array}{rrr} 5 & 1 & -1 \\ 8 & 7 & 4 \\ 9 & 7 & 6 \end{array}\right|=-1\left|\begin{array}{ll} 8 & 7 \\ 9 & 7 \end{array}\right|-4\left|\begin{array}{ll} 5 & 1 \\ 9 & 7 \end{array}\right|+6\left|\begin{array}{ll} 5 & 1 \\ 8 & 7 \end{array}\right| $$

NOTATION Write each percent as a decimal. a. \(6 \%\) b. \(4.8 \%\) c. \(13 \frac{1}{2} \%\)

For each matrix, determine the number of rows and the number of columns. a. \(\left[\begin{array}{ccc}4 & 6 & -1 \\ 1 & 9 & -3\end{array}\right]\) b. \(\left[\begin{array}{rrrr}1 & -2 & 3 & 1 \\ 0 & 1 & 6 & 4 \\ 0 & 0 & 1 & \frac{1}{3}\end{array}\right]\)

If the system\(\left\\{\begin{array}{l}4 x-3 y=7 \\ 3 x-2 y=6\end{array}\right.\)by what number should each equation be multiplied if a. the \(x\) -terms are to drop out? b. the \(y\) -terms are to drop out?

A toy company makes a total of 500 puppets in three sizes during a production run. The small puppets cost \(\$ 5\) to make and sell for \(\$ 8\) each, the standard-size puppets cost \(\$ 10\) to make and sell for \(\$ 16\) each, and the super-size puppets cost \(\$ 15\) to make and sell for \(\$ 25 .\) The total cost to make the puppets is \(\$ 4,750\) and the revenue from their sale is \(\$ 7,700 .\) How many small, standard, and super-size puppets are made during a production run?

What is the formula that finds a. Simple interest b. Distance traveled

Fill in the blanks to complete each elementary row operation: a. Type 1: Any two rows of a matrix can be ______. b. Type 2: Any row of a matrix can be ______ by a nonzero constant. c. Type 3: Any row of a matrix can be changed by _______ a nonzero constant multiple of another row to it.

Consider the system: \(\left\\{\begin{array}{l}\frac{2}{3} x-\frac{y}{6}=\frac{16}{9} \\ 0.03 x+0.02 y=0.03\end{array}\right.\) a. What algebraic step should be performed to clear the first equation of fractions? b. What algebraic step should be performed to clear the second equation of decimals?

Consider the system: \(\left\\{\begin{array}{l}-2 x+y+4 z=3 \\ x-y+2 z=1 \\\ x+y-3 z=2\end{array}\right.\) a. What is the result if equation 1 and equation 2 are added? b. What is the result if equation 2 and equation 3 are added? c. What variable was eliminated in the steps performed in parts (a) and (b)?

A dietician is to design a meal using Foods A, B, and C that will provide a patient with exactly 14 grams of fat, 13 grams of carbohydrates, and 9 grams of protein. -Each ounce of Food A contains 2 grams of fat, 3 grams of carbohydrates, and 2 grams of protein. -Each ounce of Food B contains 3 grams of fat, 2 grams of carbohydrates, and 1 gram of protein. -Each ounce of Food C contains 1 gram of fat, 1 gram of carbohydrates, and 2 grams of protein. a. Complete the following table and then form a system of three equations that could be used to determine how many ounces of each food should be used in the meal. $$ \begin{array}{|c|c|c|c|c|} \hline \begin{array}{c} \text { Name } \\ \text { of food } \end{array} & \begin{array}{c} \text { Number of } \\ \text { ounces used } \end{array} & \begin{array}{c} \text { Grams of } \\ \text { fat } \end{array} & \begin{array}{c} \text { Grams of } \\ \text { carbohydrates } \end{array} & \begin{array}{c} \text { Grams of } \\ \text { protein } \end{array} \\ \hline \mathrm{A} & a & 2 a & 3 a & \\ \mathrm{B} & b & 3 b & & b \\ \mathrm{C} & c & & c & 2 c \\ \hline \end{array} $$ Total: \(14 \quad\) Total: \(\quad\) Total: 9 b. Solve the system from part a.

Can the system \(\left\\{\begin{array}{l}2 x+5 y=7 \\ 4 x-3 y=16\end{array}\right.\)be solved more easily by the substitution or the elimination method?

For the following system, clear the equations of any fractions or decimals and write each equation in \(A x+B y+C z=D\) form. $$ \left\\{\begin{array}{l} x+y=3-4 z \\ 0.7 x-0.2 y+0.8 z=1.5 \\ \frac{x}{2}+\frac{y}{3}-\frac{z}{6}=\frac{2}{3} \end{array}\right\\{ $$

One ounce of each of three foods has the vitamin and mineral content shown in the table. How many ounces of each must be used to provide exactly 22 milligrams (mg) of niacin, 12 mg of zinc, and 20 mg of vitamin C? Milligrams per ounce in each food type $$ \begin{array}{|c|c|c|c|} \hline \text { Food } & \text { Niacin } & \text { Zinc } & \text { Vitamin C } \\ \hline \mathrm{A} & 1 \mathrm{mg} & 1 \mathrm{mg} & 2 \mathrm{mg} \\ \mathrm{B} & 2 \mathrm{mg} & 1 \mathrm{mg} & 1 \mathrm{mg} \\ \mathrm{C} & 2 \mathrm{mg} & 1 \mathrm{mg} & 2 \mathrm{mg} \\ \hline \end{array} $$

Fill in the blanks. For the system \(\left\\{\begin{array}{l}3 x+2 y=1 \\ 4 x-y=3\end{array}\right.\) \(D_{x}=-7, D_{y}=5,\) and \(D=-11\) Find the solution of the system.

a. Which matrix shown below indicates that its corresponding system of equations has no solution? b. Which matrix indicates that the equations of its corresponding system are dependent? i. \(\left[\begin{array}{lll}1 & 2 & -4 \\ 0 & 0 & 0\end{array}\right]\) ii. \(\left[\begin{array}{lll}1 & 0 & 6 \\ 0 & 1 & 0\end{array}\right]\) iii. \(\left[\begin{array}{ccc}1 & 2 & -4 \\ 0 & 0 & 2\end{array}\right]\)

Fill in the blanks. For the system \(\left\\{\begin{array}{l}2 x+3 y-z=-8 \\ x-y-z=-2 \\ -4 x+3 y+z=6\end{array}\right.\) \(D_{x}=-28, D_{y}=-14, D_{z}=14,\) and \(D=14\) Find the solution of the system.

For the following system, write each equation in standard \(A x+B y=C\) form. a. \(\left\\{\begin{array}{l}4 y=8-7 x \\ 3 x-y=2(x+4) \longrightarrow\end{array}\\{\right.\) b. For the following system, clear the equations of any fractions or decimals. $$ \left\\{\begin{array}{l} \frac{x}{5}+\frac{y}{10}=\frac{6}{5} \\ 0.3 x-0.9 y=17 \end{array}\right. $$