Chapter 6

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} x+3 y-6 z &=7 \\ 2 x-y+2 z &=0 \\ x+y+2 z &=-1 \end{aligned}$$

Solve each system by using the matrix inverse method. $$\begin{aligned} x+y-z &=6 \\ 2 x-y+z &=-9 \\ x-2 y+3 z &=1 \end{aligned}$$

Coin Collecting A coin collection made up of pennies, nickels, and quarters contains a total of 29 coins. The number of quarters is 8 less than the number of pennies. The total face value of the coins is \(\$ 1.77 .\) How many of each denomination are there

The dimensions of matrices \(A\) and \(B\) are given. Find the dimensions of the product \(A B\) and of the product BA if the products are defined. If they are not defined, say so. $$A \text { is } 7 \times 3 ; B \text { is } 2 \times 7.$$

A triangle with vertices at \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right),\) as shown in the figure, has area equal to the absolute value of \(D,\) where $$D=\frac{1}{2} \operatorname{det}\left[\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right]$$ Use \(D\) to find the area of each triangle with coordinates as given. $$P(-1,0), Q(-3,5), R(5,2)$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \leq(x+2)^{2}\\\ &y \geq-2 x^{2} \end{aligned}$$

Use a graphing calculator to solve each system. Express solutions with approximations to the nearest thousandth. $$\begin{array}{l}\sqrt{3} x-y=5 \\\100 x+y=9\end{array}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} -x+y &=-1 \\ y-z &=6 \\ x+z &=-1 \end{aligned}$$

Solve each system by using the matrix inverse method. $$\begin{aligned} x+3 y-2 z-w &=9 \\ 4 x+y+z+2 w &=2 \\ -3 x-y+z-w &=-5 \\ x-y-3 z-2 w &=2 \end{aligned}$$

Mixing Waters\(\quad\) A sparkling-water distributor wants to make up 300 gallons of sparkling water to sell for \(\$ 6.00\) per gallon. She wishes to mix three grades of water selling for \(\$ 9.00, \$ 3.00,\) and \(\$ 4.50\) per gallon, respectively. She must use twice as much of the \(\$ 4.50\) water as the \(\$ 3.00\) water. How many gallons of each should she use?

The dimensions of matrices \(A\) and \(B\) are given. Find the dimensions of the product \(A B\) and of the product BA if the products are defined. If they are not defined, say so. $$A \text { is } 4 \times 3 ; B \text { is } 2 \times 5.$$

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r} x-y < 1 \\ -1 < y < 1 \end{array}$$

Use a graphing calculator to solve each system. Express solutions with approximations to the nearest thousandth. $$\begin{aligned}&\frac{11}{3} x+y=0.5\\\&0.6 x-y=3\end{aligned}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} x+y &=1 \\ 2 x-z &=0 \\ y+2 z &=-2 \end{aligned}$$

Solve each system by using the matrix inverse method. $$\begin{aligned} 3 x+2 y-w &=0 \\ 2 x+z+2 w &=5 \\ x+2 y-z &=-2 \\ 2 x-y+z+w &=2 \end{aligned}$$

Mixing Glue\(\quad\) A glue company needs to make some glue that it can sell for \(\$ 120\) per barrel. It wants to use 150 barrels of glue worth \(\$ 100\) per barrel, along with some glue worth \(\$ 150\) per barrel and glue worth \(\$ 190\) per barrel. It must use the same number of barrels of \(\$ 150\) and \(\$ 190\) glue. How much of the \(\$ 150\) and \(\$ 190\) glue will be needed? How many barrels of \(\$ 120\) glue will be produced

The dimensions of matrices \(A\) and \(B\) are given. Find the dimensions of the product \(A B\) and of the product BA if the products are defined. If they are not defined, say so. $$A \text { is } 1 \times 6 ; B \text { is } 2 \times 4.$$

Use the concept of the area of a triangle discussed in Exercises \(39-44\) to determine whether the three points are collinear. $$(3,6),(-1,-6),(5,11)$$

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r} x+y \leq 36 \\ -4 \leq x \leq 4 \end{array}$$

Use a graphing calculator to solve each system. Express solutions with approximations to the nearest thousandth. $$\begin{aligned}&\sqrt{5} x+\sqrt[3]{6} y=9\\\&\sqrt{2} x+\sqrt[5]{9} y=12\end{aligned}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} 2 x-y+3 z &=0 \\ x+2 y-z &=5 \\ 2 y+z &=1 \end{aligned}$$

Solve each system by using the matrix inverse method. $$\begin{array}{r} x-\sqrt{2} y=2.6 \\ 0.75 x+y=-7 \end{array}$$

Three students buy different combinations of tickets for a baseball game. The first student buys 2 senior, 1 adult, and 2 student tickets for \(\$ 51 .\) The second student buys 1 adult and 5 student tickets for \(\$ 55 .\) The third student buys 2 senior, 2 adult, and 7 student tickets for \(\$ 75\). If possible, find the price of each type of ticket. Interpret your answer.

Concept Check The product \(M N\) of two matrices can be found only if the number of number of____ \(M\) equals the number of ________ of \(N.\)

Use a graphing calculator to solve each system. Express solutions with approximations to the nearest thousandth. $$\begin{aligned}&\pi x+e y=3\\\&e x+\pi y=4\end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y > x^{2}+4 x+4\\\ &y < -x^{2} \end{aligned}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} 4 x+2 y-3 z &=6 \\ x-4 y+z &=-4 \\ -x+2 z &=2 \end{aligned}$$

Solve each system by using the matrix inverse method. $$\begin{array}{l} 2.1 x+y=\sqrt{5} \\ \sqrt{2} x-2 y=5 \end{array}$$

A total of \(\$ 5000\) is invested at \(2 \%, 3 \%,\) and 4\%. The amount invested at \(4 \%\) equals the total amount invested at \(2 \%\) and \(3 \% .\) The total interest for one year is \(\$ 145 .\) If possible, find the amount invested at each interest rate. Interpret your answer.

Concept Check In finding the product \(A B\) of matrices \(A\) and \(B,\) the first row, second column, entry is found by multiplying the ______ elements in \(A\) and the ______ elements in \(B\) and then ______ these products.

Use the concept of the area of a triangle discussed in Exercises \(39-44\) to determine whether the three points are collinear. $$(4,-5),(-2,10),(6,-10)$$

Explain how one can determine whether a system is inconsistent or has dependent equations when using the substitution or elimination method.

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \geq(x-2)^{2}+3\\\ &y \leq-(x-1)^{2}+6 \end{aligned}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} x+y-2 z &=-6 \\ x-y+z &=4 \\ 2 x-z &=-1 \end{aligned}$$

Triangle Dimensions The sum of the measures of the angles of any triangle is \(180^{\circ} .\) In a certain triangle, the Targest angle measures \(55^{\circ}\) less than twice the medium angle, and the smallest measures \(25^{\circ}\) less than the medium angle. Find the measures of the three angles.

Solve each system by using the matrix inverse method. $$\begin{aligned} \pi x+e y+\sqrt{2} z &=1 \\ e x+\pi y+\sqrt{2} z &=2 \\ \sqrt{2} x+e y+\pi z &=3 \end{aligned}$$

If possible, find \(A B\) and \(B A.\) $$A=\left[\begin{array}{rr} 1 & -1 \\ 2 & 0 \end{array}\right], \quad B=\left[\begin{array}{rr} -2 & 3 \\ 1 & 2 \end{array}\right]$$

For what value(s) of \(k\) will the following system of linear equations have no solution? infinitely many solutions? $$\begin{array}{r}x-2 y=3 \\\\-2 x+4 y=k\end{array}$$

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r} x \geq 0 \\ x+y \leq 4 \\ 2 x+y \leq 5 \end{array}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} 2 x+y-3 z &=1 \\ x+y+2 z &=5 \\ 3 x+2 y-z &=-3 \end{aligned}$$

Triangle Dimensions The perimeter of a triangle is 59 inches. The longest side is 11 inches longer than the medium side, and the medium side is 3 inches more than the shortest side. Find the length of each side.

Solve each system by using the matrix inverse method. $$\begin{aligned} (\log 2) x+(\ln 3) y+(\ln 4) z &=1 \\ (\ln 3) x+(\log 2) y+(\ln 8) z &=5 \\ (\log 12) x+(\ln 4) y+(\ln 8) z &=9 \end{aligned}$$

If possible, find \(A B\) and \(B A.\) $$A=\left[\begin{array}{rr} -3 & 5 \\ 2 & 7 \end{array}\right], \quad B=\left[\begin{array}{rr} -1 & 2 \\ 0 & 7 \end{array}\right]$$

Use the concept of the area of a triangle discussed in Exercises \(39-44\) to determine whether the three points are collinear. $$(-1,-4),(3,8),(6,17)$$

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A line and a circle; no points.

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} 3 x-2 y & \geq 6 \\ x+y & \leq-5 \\ y & \leq 4 \end{aligned}$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{array}{c} 0.07 x+0.23 y=9 \\ -1.25 x+0.33 y=2.4 \end{array}$$

Investment Decisions \(\mathrm{A}\) student invested \(\$ 10,000\) in three parts. With one part, she bought mutual funds that offered a return of \(4 \%\) per year. The second part, which amounted to twice the first, was used to buy government bonds paying \(4.5 \%\) per year. She put the rest into a savings account that paid \(2.5 \%\) annual interest. During the first year, the total interest was \(\$ 415 .\) How much did she invest at each rate?

Use the determinant theorems to find each determinant. $$\operatorname{det}\left[\begin{array}{lll}1 & 0 & 0 \\\1 & 0 & 1 \\\3 & 0 & 0\end{array}\right]$$

If possible, find \(A B\) and \(B A.\) $$A=\left[\begin{array}{rrr} 5 & -7 & 2 \\ 0 & 1 & 5 \end{array}\right], \quad B=\left[\begin{array}{rrr} 9 & 8 & 7 \\ 1 & -1 & -2 \end{array}\right]$$