Exam 4: Systems of Equations and Inequalities

The sum of three positive numbers is $65$ . The second number is 5 $\text { less }$ than the first, and the third is 5 times the first. What is the $\text { third }$ number?

Solve the system of linear equations below by the method of elimination, if a single solution exists. $\left\{ \begin{array} { c } - x + 2 y = 3 \\- 2 x + 4 y = 8\end{array} \right.$

Solve the system of linear equations below. $\left\{ \begin{array} { c } 4 x + 3 y + z = 6 \\5 x - 4 y - 2 z = - 4 \\2 x + 5 y - 3 z = 38\end{array} \right.$

Solve the system of linear equations below by the method of elimination, if a single solution exists. $\left\{ \begin{array} { l } - 4 x + y = - 4 \\12 x - 3 y = 3\end{array} \right.$

Describe the elementary row operation used to transform the first matrix $\left[ \begin{array} { c c c } 3 & 5 & 6 \\0 & 2 & 2 \\0 & - 3 & - 4\end{array} \right]$ into the second matrix $\left[ \begin{array} { l l l } 3 & 5 & 6 \\0 & 2 & 2 \\0 & 3 & 2\end{array} \right]$ .

Solve for x in the matrix below by using elementary row operations to form a row-equivalent matrix. $\left[ \begin{array} { c c c c } 6 & 3 & - 2 & 0 \\- 2 & - 3 & - 3 & - 9 \\4 & 0 & - 3 & 8\end{array} \right]$ $\left[ \begin{array} { c c c c } 6 & 3 & - 2 & 0 \\- 2 & - 3 & - 3 & - 9 \\- 2 & - 3 & x & y\end{array} \right]$

Determine the value of k such that the system of linear equations below is inconsistent. $\left\{ \begin{array} { c } 7 x + 18 y = - 3 \\5 x + k y = 3\end{array} \right.$

Graph the system of linear inequalities below. $\left\{ \begin{array} { l } x + y \geq - 2 \\x - y \leq - 1\end{array} \right.$

Solve the system of equations below by the method of substitution. $\left\{ \begin{array} { l } 9 x + 5 y = - 66 \\5 x + 9 y = - 74\end{array} \right.$

A person plans to invest up to $10,000 in two different interest-bearing accounts, account X and account Y. Account Y is to contain at least $ $1000$ . Moreover, account X should have at least twice the amount in account Y. Graph the system of linear inequalities describing the various amounts that can be deposited in each account.

Use matrices to solve the system of linear equations below. $\left\{ \begin{array} { l } - 2 x - y = - 13 \\x + 2 y = 11\end{array} \right.$

Determine whether the system is consistent or inconsistent. $\left\{\begin{array}{l} x+y=1 \\ 2 x+2 y=6\end{array}\right.$

Use the graph of the equation $\left\{ \begin{array} { l } 8 x - 5 y = 14 \\3 x + 5 y = 8\end{array} \right.$ to determine whether the system has any solutions. Find any solutions that exist.

Evaluate the determinant of the matrix. Expand by minors along the row or column that appears to make the computation easiest. $\left[ \begin{array} { c c c } 1 & 0 & 4 \\1 & - 4 & 2 \\- 1 & 2 & 3\end{array} \right]$

Solve the system of linear equations below. $\left\{ \begin{array} { r } x + y - 2 z = - 8 \\x - y - 3 z = - 5 \\2 x + 4 z = - 4\end{array} \right.$

Solve the system of linear equations below by the method of elimination. $\left\{ \begin{array} { c } 4 r - 5 s = 5 \\2 r - 5 s = 15\end{array} \right.$

A total of $\$ 15,000$ is invested in two bonds that pay $5.5 \%$ and $6 \%$ simple interest. The annual interest is $\$ 865$ . How much is invested in the $6 \%$ bond?

Find two positive integers that satisfy the requirements that the difference of four times the smaller number and the larger number is 14 and the sum of the smaller number and four times the larger number is 335 .

A grocer wishes to mix three kinds of nuts to obtain $40$ pounds of a mixture priced at $\$ 4.45$ per pound. Peanuts cost $\$ 3$ per pound, almonds cost $\$ 5$ per pound, and pistachios cost $\$ 5.5$ per pound. Half of the mixture is composed of peanuts and almonds. How many pounds of $\text { peanuts }$ should the grocer use?

Form the coefficient matrix for the system of linear equations below. $\left\{ \begin{array} { r } 6 x + 3 y + 7 z = 0 \\x + 6 y + z = 7 \\8 x + 9 y + 7 z = - 9\end{array} \right.$