Exam 4: Systems of Equations and Inequalities

Use the graph of the equation $\left\{ \begin{array} { l } 4 x - y = 8 \\- 8 x + 2 y = - 12\end{array} \right.$ to determine whether the system has any solutions. Find any solutions that exist.

Solve the system of linear equations below by the method of elimination. $\left\{ \begin{array} { l } 9 x + 4 y = - 108 \\- 7 x + 2 y = 38\end{array} \right.$

Evaluate the determinant of the matrix. $\left[ \begin{array} { c c c } - 0.7 & - 0.6 & - 0.9 \\- 0.9 & - 0.5 & 0.5 \\1 & - 9 & 3\end{array} \right]$

Write the system of linear equations represented by the augmented matrix. Then use back-substitution to find the solution. Use variables x , y , and z . $\left[ \begin{array} { l l l : l } 1 & 2 & 1 & - 5 \\0 & 1 & 2 & - 6 \\0 & 0 & 1 & - 1\end{array} \right]$

The slope and y -intercept of the line $y = m x + b$ that best fits the three noncollinear points $( 2,2 ) , ( 3,3 )$ and $( 5,4 )$ are given by the solution of the following system of linear equations. $\left\{ \begin{array} { c } 77 m + 12 b = 46 \\12 m + 5 b = 8\end{array} \right.$ Solve the system and find the equation of the best-fitting line. Round parameters to the nearest thousandth.

A furniture company can sell all the tables and chairs it produces. Each table requires $1.5$ hours in the assembly center and $2.25$ hours in the finishing center. Each chair requires $1$ hour in the assembly center and $0.5$ hour in the finishing center. The company s assembly center is available 9 hours per day, and its finishing center is available 12 hours per day. Graph the system of linear inequalities describing the different production levels, where t is the number of tables and c is the number of chairs.

Determine the order of the given matrix below. $\left[ \begin{array} { c } - 4 \\- 1 \\0 \\4\end{array} \right]$

You are setting up a small business and have made an initial investment of $\$ 18,000$ . The unit cost of the guitar you are producing is $37.5 , and the selling price is $77.5 . How many guitars must you sell to break even?

Solve the system of linear equations below by the method of elimination, if a single solution exists. $\left\{ \begin{array} { c } - 9 u - 8 v = - 8 \\9 u + 2 v = 2\end{array} \right.$

Use a determinant to find the equation of the line through the points $( 2.3,7.6 )$ and $( 1.3 , - 1.4 )$ .

Solve the system of linear equations below. $\left\{ \begin{aligned}2 x + 3 y & = - 18 \\z & = 4 \\x - 8 y - z & = 6\end{aligned} \right.$

$\text { Sixteen }$ pounds of mixed birdseed sells for $7.2 per pound. The mixture is obtained from two kinds of birdseed, with one variety priced at $6.2 per pound and the other at $7.8 per pound. How many pounds of each variety of birdseed are used in the mixture?

Graph the system of linear inequalities below. $\left\{ \begin{array} { l } x + y < 2 \\6 x + 2 y > 4\end{array} \right.$

Decide whether the system $\left\{ \begin{array} { c } 5 x - 6 y = 4 \\- 15 x + 18 y = 14\end{array} \right.$ is consistent or inconsistent.

14 pounds of mixed nuts sells for $6.21 per pound. The mixture is obtained from two kinds of nuts, peanuts priced at $5.5 per pound and cashews at $6.5 per pound. How many pounds of each variety of nut are used in the mixture?

Graph the system of linear inequalities below. $\left\{ \begin{array} { l } 2 x + y \leq 2 \\x \geq - 1 \\y \geq 0\end{array} \right.$

Determine which of the following ordered pairs, $( - 1.5 , - 1.5 )$ , $( - 0.5,9 )$ , $( 5.5 , - 3 )$ , $( 4.5,1 )$ , or $( 1.5,3.5 )$ is a solution to the system of equations below. $\left\{ \begin{array} { l } 6 x - 9 y = 4.5 \\x + 3 y = - 6\end{array} \right.$

Solve the system of equations below by the method of substitution. $\left\{ \begin{array} { l } \frac { 3 } { 5 } x - \frac { 3 } { 2 } y = - \frac { 21 } { 10 } \\\\\frac { 5 } { 2 } x + \frac { 2 } { 7 } y = \frac { 335 } { 14 }\end{array} \right.$

Solve the system of linear equations below. $\left\{ \begin{array} { c } 2 x - 3 y - 6 z = - 34 \\4 x - y + 5 z = 17 \\3 x + 6 z = 24\end{array} \right.$

Find the area of a quadrilateral with vertices (7,-6) , (3,-2) , (8,-4) , and (-4,-5) .