91Ó°ÊÓ

Assume that the matrix is the augmented matrix of a system of linear equations, and (a) determine the number of equations and the number of variables, and (b) find the value(s) of \(k\) such that the system is consistent. Then assume that the matrix is the coefficient matrix of a homogeneous system of linear equations, and repeat parts (a) and (b). $$ A=\left[\begin{array}{rrr} 1 & k & 2 \\ -3 & 4 & 1 \end{array}\right] $$

Assume that the matrix is the augmented matrix of a system of linear equations, and (a) determine the number of equations and the number of variables, and (b) find the value(s) of \(k\) such that the system is consistent. Then assume that the matrix is the coefficient matrix of a homogeneous system of linear equations, and repeat parts (a) and (b). $$ A=\left[\begin{array}{rrr} 2 & -1 & 3 \\ -4 & 2 & k \\ 4 & -2 & 6 \end{array}\right] $$

Find values of \(a, b,\) and \(c\) (if possible) such that the system of linear equations has (a) a unique solution, (b) no solution, and (c) infinitely many solutions. $$ \begin{aligned} x+y &=2 \\ y+z &=2 \\ x+z &=2 \\ a x+b y+c z &=0 \end{aligned} $$

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) A \(4 \times 7\) matrix has four columns. (b) Every matrix has a unique reduced row-echelon form. (c) A homogeneous system of four linear equations in four variables is always consistent. (d) Multiplying a row of a matrix by a constant is one of the elementary row operations.

State why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. $$\begin{aligned}&16 x+3 y+z=0\\\&16 x+2 y-z=0\end{aligned}$$

Nutrition One eight-ounce glass of apple juice and one eight-ounce glass of orange juice contain a total of 227 milligrams of vitamin C. Two eight-ounce glasses of apple juice and three eight-ounce glasses of orange juice contain a total of 578 milligrams of vitamin \(C\). How much vitamin \(\mathrm{C}\) is in an eight-ounce glass of each type of juice?

Airplane Speed Two planes start from Los Angeles International Airport and fly in opposite directions. The second plane starts \(\frac{1}{2}\) hour after the first plane, but its speed is 80 kilometers per hour faster. Two hours after the first plane departs, the planes are 3200 kilometers apart. Find the airspeed of each plane.

True or False? In Exercises 69 and \(70,\) determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) A linear system can have exactly two solutions. (b) Two systems of linear equations are equivalent when they have the same solution set. (c) A system of three linear equations in two variables is always inconsistent.

CAPSTONE Find values of \(a, b,\) and \(c\) such that the system of linear equations has (a) exactly one solution, (b) infinitely many solutions, and (c) no solution. Explain. $$\begin{array}{r}x+5 y+z=0 \\\x+6 y-z=0 \\\2 x+a y+b z=c\end{array}$$

The graphs of the two equations appear to be parallel. Solve the system of equations algebraically. Explain why the graphs are misleading. (GRAPHS CANNOT COPY?) $$\begin{array}{rr}100 y-x= & 200 \\\99 y-x= & -198\end{array}$$