Exam 6: Systems of Linear Equations and Matrices

Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system. - $3 x-\frac{5 y}{7}=10$ $\frac{2 x}{3}-\frac{9 y}{7}=\frac{19}{5}$

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,dependent, or inconsistent. - $x+y+z=7$ $x-y+2 z=7$ $2 x+3 z=14$

Find the inverse, if it exists, of the given matrix. - $\mathrm{A}=\left[\begin{array}{rr}0 & 5 \\ -5 & 2\end{array}\right]$

Determine whether the given ordered set of numbers is a solution of the system of equations. - $(-6,-3)$ $x+y=3$ $\mathrm{x}-\mathrm{y}=9$

Find the order of the matrix product $\mathrm{AB}$ and the product $\mathrm{BA}$ , whenever the products exist. - $A$ is $1 \times 3$ , B is $1 \times 3$ .

Use the Gauss-Jordan method to solve the system of equations. - $-3 x-y-9 z=-75$ $3 x+5 y-2 z=44$ $-8 \mathrm{x}-6 \mathrm{y}+\mathrm{z}=-95$

Solve the matrix equation for $X$ .} - $\mathrm{A}=\left[\begin{array}{rr}-1 & 3 \\ 4 & 2\end{array}\right], \mathrm{B}=\left[\begin{array}{rr}-1 & 9 \\ -10 & 6\end{array}\right], \mathrm{AX}=\mathrm{B}$

Obtain an equivalent system by performing the stated elementary operation on the system. -Multiply the third equation by $1 / 8$ . $7 x-y+8 z+w=-10$ $8 \mathrm{x} \quad-\mathrm{z}-2 \mathrm{w}=7$ $8 x-16 y+7 z-w=-24$ $\mathrm{x}+3 \mathrm{y}-8 \mathrm{z}=1$

Solve the problem by writing and solving a suitable system of equations. -Julia is preparing a meal by combining three ingredients. One unit of each ingredient provides the following quantities (in grams) of carbohydrates, fat, and protein. Ideally the meal should contain 25 grams of protein, 35 grams of carbohydrates, and 13 grams of fat. How many units of each ingredient should Julia use?

Write the system of equations associated with the augmented matrix. Do not solve. - $\left[\begin{array}{rr|r}1 & 0 & -10 \\ 0 & 1 & 9\end{array}\right]$

Perform the row operations on the matrix and write the resulting matrix. -Replace $R_{2}$ by $\frac{1}{3} R_{1}+\frac{1}{2} R_{2}$ $\left[\begin{array}{rr|r}3 & 0 & 9 \\ -2 & 4 & 8\end{array}\right]$

Solve the problem. -Su ppose the following matrix represents the input-output matrix of a primitive economy. How much of each commodity should be produced to produce 15 bushels of yams and 4 pigs?

Use the Gauss-Jordan method to solve the system of equations. - $x-y+5 z=-17$ $5 \mathrm{x}+\mathrm{z}=-3$ $x+4 y+z=5$

The diagram shows the roads connecting four cities. The matrix A below represents the number of routes between each pair of cities without passing through another city. Calculate $\mathrm{A}^{3}$ . What information is given by the entry in row 2 , column 3 of $\mathrm{A}^{3}$ ?

Perform the indicated operation where possible. - $\left[\begin{array}{rr}-1 & 5 \\ 0 & 4 \\ 8 & -4\end{array}\right]-\left[\begin{array}{ll}2 & 1 \\ 7 & 4 \\ 3 & 2\end{array}\right]$

Write the word or phrase that best completes each statement or answers thequestion. -A simplified economy has only two industries, the electric company and the gas company. Each dollar's worth of the electric company's output requires 0.20 of its own output and 0.4 of the gas company's output. Each dollar's worth of the gas company's output requires 0.50 of its own output and 0.7 of the electric company's output. What should the production of electricity and gas be (in dollars) if there is a $\$ 14$ million demand for electricity and a $\$ 20$ million demand for gas?

Write a system of equations and use the inverse of the coefficient matrix to solve the system. -A bakery sells three types of cakes, each requiring the amount of ingredients shown. To fill its orders for these cakes, the bakery used 72 cups of flour, 48 cups of sugar, and 54 eggs. How many cakes of each type were made?

Find the inverse, if it exists, of the given matrix. - $\mathrm{A}=\left[\begin{array}{rr}-4 & 2 \\ 2 & -1\end{array}\right]$

Perform the indicated operation where possible. - $\left[\begin{array}{r}3 \\ -1 \\ -4\end{array}\right]+\left[\begin{array}{r}-5 \\ 4 \\ 6\end{array}\right]$

Solve the matrix equation for $X$ .} - $\mathrm{A}=\left[\begin{array}{rr}5 & 12 \\ 2 & 5\end{array}\right], \mathrm{B}=\left[\begin{array}{r}-3 \\ 5\end{array}\right], \mathrm{AX}=\mathrm{B}$