Chapter 6

( Refer to Examples 3-5.) LetA be the given matrix. Find \(A^{-1}\) without a calculator. $$ \left[\begin{array}{rrr} 2 & -2 & 1 \\ 1 & 3 & 2 \\ 4 & -2 & 4 \end{array}\right] $$

Use Cramer's rule to solve the system of linear equations. $$ \begin{array}{l} 2 x+y=-3 \\ -4 x-6 y=-7 \end{array} $$

Perform each row operation on the given matrix by completing the matrix at the right. $$ \left[\begin{array}{rrr|r} 1 & -2 & 3 & 6 \\ 2 & 1 & 4 & 5 \\ -3 & 5 & 3 & 2 \end{array}\right] \begin{array}{r} R_{2}-2 R_{1} \rightarrow \\ R_{3}+3 R_{1} \rightarrow \end{array}\left[\begin{array}{rrr|r} 1 & -2 & 3 & 6 \\ & & & \end{array}\right] $$

If possible, solve the system. $$ \begin{array}{rr} x+2 y+z= & 0 \\ 3 x+2 y-z= & 4 \\ -x+2 y+3 z= & -4 \end{array} $$

Graph the solution set to the system of inequalities. $$ \begin{aligned} &x-2 y \geq 0\\\ &x-3 y \leq 3 \end{aligned} $$

Use Cramer's rule to solve the system of linear equations. $$ \begin{array}{r} -2 x+3 y=8 \\ 4 x-5 y=3 \end{array} $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{array}{r} x+2 y=3 \\ -x-y=7 \end{array} $$

Write a system of linear equations with two variables whose solution satisfies the problem. State what each variable represents. Then solve the system. The screen of a rectangular television set is 2 inches wider than it is high. If the perimeter of the screen is 38 inches, find its dimensions.

If possible, solve the system. $$ \begin{array}{rr} -x+\quad 2 z= -9 \\ y+4 z= -13 \\ 3 x+y \quad =13 \end{array} $$

Graph the solution set to the system of inequalities. $$ \begin{array}{l} 2 x-4 y \geq 4 \\ x+y \leq 0 \end{array} $$

Use Cramer's rule to solve the system of linear equations. $$ \begin{array}{r} 5 x-3 y=4 \\ -3 x-7 y=5 \end{array} $$

( Refer to Examples 3-5.) LetA be the given matrix. Find \(A^{-1}\) without a calculator. $$ \left[\begin{array}{rrr} 1 & -1 & 1 \\ -1 & 2 & 1 \\ 0 & 2 & 1 \end{array}\right] $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{aligned} 2 x+4 y &=10 \\ x-2 y &=-3 \end{aligned} $$

Write a system of linear equations with two variables whose solution satisfies the problem. State what each variable represents. Then solve the system. The sum of two numbers is 300 and their difference is \(8 .\) Find the two numbers.

If possible, solve the system. $$ \begin{aligned} x+y+z &=-1 \\ 2 x+\quad z &=-6 \\ 2 y+3 z &=0 \end{aligned} $$

Graph the solution set to the system of inequalities. $$ \begin{array}{r} x^{2}+y^{2} \leq 4 \\ y \geq 1 \end{array} $$

LetA be the given matrix. Find \(A^{-1}\). $$ \left[\begin{array}{ll} 0.5 & -1.5 \\ 0.2 & -0.5 \end{array}\right] $$

Use Cramer's rule to solve the system of linear equations. $$ \begin{array}{r} 7 x+4 y=23 \\ 11 x-5 y=70 \end{array} $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{array}{rr} x+2 y+z= & 3 \\ x+y-z= & 3 \\ -x-2 y+z= & -5 \end{array} $$

Write a system of linear equations with two variables whose solution satisfies the problem. State what each variable represents. Then solve the system. Admission prices to a movie are \(\$ 4\) for children and \(\$ 7\) for adults. If 75 tickets were sold for \(\$ 456\) how many of each type of ticket were sold?

If possible, solve the system. $$ \begin{aligned} \frac{1}{2} x-y+\frac{1}{2} z &=-4 \\ x+2 y-3 z &=20 \\ -\frac{1}{2} x+3 y+2 z &=0 \end{aligned} $$

Graph the solution set to the system of inequalities. $$ \begin{aligned} &x^{2}-y \leq 0\\\ &x^{2}+y^{2} \leq 6 \end{aligned} $$

Use Cramer's rule to solve the system of linear equations. $$ \begin{array}{rr} -7 x+5 y= & 8.2 \\ 6 x+4 y= & -0.4 \end{array} $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{array}{r} x+y+z=6 \\ 2 x+3 y-z=3 \\ x+y+2 z=10 \end{array} $$

Write a system of linear equations with two variables whose solution satisfies the problem. State what each variable represents. Then solve the system. A sample of 16 dimes and quarters has a value of \(\$ 2.65 .\) How many of each type of coin are there?

If possible, solve the system. $$ \begin{aligned} \frac{3}{4} x+y+\frac{1}{2} z &=-3 \\ x+y-z &=-8 \\ \frac{1}{4} x-2 y+z &=-4 \end{aligned} $$

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

Graph the solution set to the system of inequalities. $$ \begin{array}{r} 2 x^{2}+y \leq 0 \\ x^{2}-y \leq 3 \end{array} $$

Use Cramer's rule to solve the system of linear equations. $$ \begin{array}{rr} 1.7 x-2.5 y= & -0.91 \\ -0.4 x+0.9 y= & 0.423 \end{array} $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{array}{rr} x+2 y-z= & -1 \\ 2 x-y+z= & 0 \\ -x-y+2 z= & 7 \end{array} $$

Five hundred tickets were sold for a play, generating \(\$ 3560 .\) The prices of the tickets were \(\$ 5\) for children, \(\$ 7\) for students, and \(\$ 10\) for adults. There were 180 more student tickets sold than adult tickets. Find the number of each type of ticket sold.

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

Graph the solution set to the system of inequalities. $$ \begin{aligned} &x^{2}+2 y \leq 4\\\ &x^{2}-y \leq 0 \end{aligned} $$

Use Cramer's rule to solve the system of linear equations. $$ \begin{aligned} -2.7 x+1.5 y &=-1.53 \\ 1.8 x-5.5 y &=-1.68 \end{aligned} $$

LetA be the given matrix. Find \(A^{-1}\). $$ \left[\begin{array}{rrr} -2 & 0 & 1 \\ 5 & -4 & 1 \\ 1 & -2 & 0 \end{array}\right] $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{aligned} x+3 y-2 z &=-4 \\ 2 x+6 y+z &=-3 \\ x+y-4 z &=-2 \end{aligned} $$

One thousand tickets were sold for a baseball game. There were one hundred more adult tickets sold than student tickets, and there were four times as many tickets sold to students as to children. How many of each type of ticket were sold?

Graph the solution set to the system of inequalities. $$ \begin{aligned} &x^{2}+y^{2} \leq 4\\\ &x^{2}+2 y \leq 2 \end{aligned} $$

LetA be the given matrix. Find \(A^{-1}\). $$ \left[\begin{array}{rrr} 2 & -2 & 1 \\ 0 & 5 & 8 \\ 0 & 0 & -1 \end{array}\right] $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{aligned} 3 x+y+3 z &=14 \\ x+y+z &=6 \\ -2 x-2 y+3 z &=-7 \end{aligned} $$

Three students buy lunch in the cafeteria. One student buys 2 hamburgers, 2 orders of fries, and 1 soda for \(\$ 9 .\) Another student buys 1 hamburger, 1 order of fries, and 1 soda for \(\$ 5 .\) The third student buys 1 hamburger and 1 order of fries for \(\$ 5 .\) If possible, find the cost of each item. Interpret the results.

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

Graph the solution set to the system of inequalities. $$ \begin{aligned} &2 x+3 y \leq 6\\\ &\frac{1}{2} x^{2}-y \leq 2 \end{aligned} $$

LetA be the given matrix. Find \(A^{-1}\). $$ \left[\begin{array}{rrr} 2 & 0 & 2 \\ 1 & 5 & 0 \\ -1 & 0 & 2 \end{array}\right] $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{array}{rr} x+3 y-2 z= & 3 \\ -x-2 y+z= & -2 \\ 2 x-7 y+z= & 1 \end{array} $$

The table shows the total cost of purchasing various combinations of differently priced CDs. The types of CDs are labeled \(A, B,\) and \(C\). $$ \begin{aligned} &\begin{array}{cccc} \mathbf{A} & \mathbf{B} & \mathbf{C} & \text { Total Cost } \\ \hline 2 & 1 & 1 & \$ 48 \\ \mathbf{3} & \mathbf{2} & \mathbf{1} & \mathbf{\$ 7 1} \\ \mathbf{1} & 1 & \mathbf{2} & \mathbf{\$ 5 3} \end{array}\\\ \end{aligned} $$ (a) Let \(a\) be the cost of a CD of type \(A, b\) be the cost of a CD of type \(B\), and \(c\) be the cost of a CD of type C. Write a system of three linear equations whose solution gives the cost of each type of CD. (b) Solve the system of equations and check your answer.

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rr}3 & -1 \\\1 & 0 \\\\-2 & -4\end{array}\right], \quad B=\left[\begin{array}{rrr}-2 & 5 & -3 \\\9 & -7 & 0\end{array}\right]$$

LetA be the given matrix. Find \(A^{-1}\). $$ \left[\begin{array}{rrr} 3 & -1 & -1 \\ -1 & 3 & -1 \\ -1 & -1 & 3 \end{array}\right] $$

Use Gaussian elimination with backward substitution to solve the system of linear equations. Write the solution as an ordered pair or an ordered triple whenever possible. $$ \begin{aligned} 2 x+5 y+z &=8 \\ x+2 y-z &=2 \\ 3 x+7 y &=5 \end{aligned} $$

Geometry The largest angle in a triangle is \(25^{\circ}\) more than the smallest angle. The sum of the measures of the two smaller angles is \(30^{\circ}\) more than the measure of the largest angle. (a) Let \(x, y,\) and \(z\) be the measures of the three angles from largest to smallest. Write a system of three linear equations whose solution gives the measure of each angle. (b) Solve the system of equations and check your answer.