Chapter 7

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

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 cqual 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 coondinates as given. (GRAPH CAN'T COPY) $$P(-1,0), Q(-3,5), R(5,2)$$

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

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. $$A \text { is } 7 \times 3 ; B \text { is } 2 \times 7$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{array}{r} 2 x+y+3 z=4 \\ -3 x-y-4 z=5 \\ x+y+2 z=0 \end{array}$$

Solve each system by elimination. $$\begin{aligned}&\frac{x}{5}+3 y=31\\\&2 x-\frac{y}{5}=8\end{aligned}$$

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

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

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. $$A \text { is } 4 \times 3 ; B \text { is } 2 \times 5$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{array}{l} x+y+z=0 \\ x-y-z=3 \\ x+3 y+3 z=5 \end{array}$$

Solve each system by elimination. $$\begin{array}{l}\frac{2 x-1}{3}+\frac{y+2}{4}=4 \\\\\frac{x+3}{2}-\frac{x-y}{3}=3\end{array}$$

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

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

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. $$A \text { is } 1 \times 6 ; B \text { is } 2 \times 4$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{array}{r} x+3 y+z=6 \\ 3 x+y-z=6 \\ x-y-z=0 \end{array}$$

Solve each system by elimination. $$\begin{aligned}&\frac{x+6}{5}+\frac{2 y-x}{10}=1\\\&\frac{x+2}{4}+\frac{3 y+2}{5}=-3\end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{array}{c}x+y \geq 0 \\\2 x-y \geq 3\end{array}$$

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

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 analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 2 x-y+2 z &=6 \\ -x+y+z &=0 \\ -x-3 z &=-6 \end{aligned}$$

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

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

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

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 analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} x+2 y+z &=0 \\ 3 x+2 y-z &=4 \\ -x+2 y+3 z &=-4 \end{aligned}$$

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

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r}2 x+y>2 \\\x-3 y<6\end{array}$$

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

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

Solve the system from $$\begin{aligned} 25 x+40 y+20 z &=2200 \\ 4 x+2 y+3 z &=280 \\ 3 x+2 y+z &=180 \end{aligned}$$

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

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

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r}4 x+3 y<12 \\\y+4 x>-4\end{array}$$.

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

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{array}{l} 3 x-2 y+z=2 \\ x+2 y-2 z=1 \\ x-4 y+2 z=-1 \end{array}$$

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

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

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r}3 x+5 y \leq 15 \\\x-3 y \geq 9\end{array}$$

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

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} x-y+4 z &=0 \\ 2 x+2 y-2 z &=-1 \\ -x-y+2 z &=1 \end{aligned}$$

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?

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. $$\begin{aligned}&y \leq x\\\&x^{2}+y^{2}<1\end{aligned}$$

Use the determinant theorems to find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}-1 & 2 & 4 \\\4 & -8 & -16 \\\3 & 0 & 5\end{array}\right]$$

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} x-2 y+z &=4 \\ 2 x+2 y-2 z &=-1 \\ x+4 y+z &=7 \end{aligned}$$

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 S190 glue. How much of the \(\$ 150\) and \(\$ 190\) glue will be needed? How many barrels of \(\$ 120\) glue will be produced

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

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

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}4 x-3 y & \leq 12 \\\y & \leq x^{2}\end{aligned}$$