Chapter 7

In the process called _____, you find the maximum or minimum value of a quantity.

The process of writing a rational expression as the sum of two or more simpler rational expressions is called_____ _____ ______.

The first step in solving a system of equations by the method of _____ is to obtain coefficients for \(x\) (or y ) that differ only in sign.

A_____________of a system of equations is an ordered pair that satisfies each equation in the system.

The _________ of an inequality is the collection of all solutions of the inequality.

Two systems of equations that have the same solution set are called _____ systems.

If the degree of the numerator of a rational expression is greater than or equal to the degree of the denominator, then the fraction is called ______.

The first step in solving a system of equations by the method of____________________is to solve one of the equations for one variable in terms of the other variable.

A solution of a system of three linear equations in three unknowns can be written as an _____, which has the form \(( x , y , z )\)

The _____ function of a linear programming problem gives the quantity to be maximized or minimized.

A ____________________ of a system of inequalities in \(x\) and \(y\) is a point \((x, y)\) that satisties each inequality in the system.

A system of linear equations that has at least one solution is called _____ ,whereas a system of linear equations that has no solution is called ____.

Each fraction on the right side of the equation \(\frac{x-1}{x^{2}-8 x+15}=\frac{-1}{x-3}+\frac{2}{x-5}\) is a ______ _______.

Graphically, the solution of a system of two equations is the_______________of__________________of the graphs of the two equations.

The process used to write a system of linear equations in row-echelon form is called _______ elimination.

In business applications, the _____ _____, is defined as the price \(p\) and the number of units \(x\) that satisfy both the demand and supply equations

A ______________ of a system of inequalities in two variables is the region common to the graphs of every inequality in the system.

You obtain the _______ ______ after multiplying each side of the partial fraction decomposition form by the least common denominator.

In business applications, the point at which the revenue equals costs is called the____________point.

Interchanging two equations of a system of linear equations is a ____ ____ that produces an equivalent system.

Solving a System by Elimination In Exercises \(5-12,\) solve the system by the method of elimination. Label each line with its equation. To print an enlarged copy of the graph, go to Math Graphs.com. $$ \left\\{\begin{array}{r}{2 x+y=5} \\ {x-y=1}\end{array}\right. $$

In Exercises 5-18, sketch the graph of the inequality. $$y<5-x^{2}$$

Matching. Match the rational expression with the form of its decomposition. IThe decompositions are labeled (a), (b), \((\mathrm{c}),\) and (d).] \(\begin{array}{ll}{\text { (a) } \frac{A}{x}+\frac{B}{x+2}+\frac{C}{x-2}} & {\text { (b) } \frac{A}{x}+\frac{B}{x-4}} \\ {\text { (c) } \frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x-4}} & {\text { (d) } \frac{A}{x}+\frac{B x+C}{x^{2}+4}}\end{array}\) $$\frac{3 x-1}{x(x-4)}$$

A system of equations is called _______ when the number of equations differs from the number of variables in the system.

If a linear programming problem has a solution, then it must occur at a _____ of the set of feasible solutions.

In Exercises 5-18, sketch the graph of the inequality. $$y^{2}-x<0$$

Solving a System by Elimination In Exercises \(5-12,\) solve the system by the method of elimination. Label each line with its equation. To print an enlarged copy of the graph, go to MathGraphs.com. $$\left\\{\begin{aligned} x+3 y &=1 \\\\-x+2 y &=4 \end{aligned}\right.$$

Checking Solutions In Exercises 5 and \(6,\) determine whether each ordered pair is a solution of the system. $$\left\\{\begin{array}{r}{4 x^{2}+y=3} \\ {-x-y=11}\end{array}\right.$$ $$\begin{array}{ll}{\text { (a) }(2,-13)} & {\text { (b) }(2,-9)} \\ {\text { (c) }\left(-\frac{3}{2},-\frac{31}{3}\right)} & {\text { (d) }\left(-\frac{7}{4},-\frac{37}{4}\right)}\end{array}$$

The equation $$s = \frac { 1 } { 2 } a t ^ { 2 } + v _ { 0 } t + s _ { 0 }$$ is called the ______ equation, and it models the height \(s\) of an object at time \(t\) that is moving in a vertical line with a constant acceleration a.

Solving a Linear Programming Problem, find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) $$ \begin{array}{c}{\text { Objective function: }} \\ {z=4 x+3 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {x+y \leq 5}\end{array} $$

In Exercises 5-18, sketch the graph of the inequality. $$x \geq 6$$

Solving a System by Elimination In Exercises \(5-12,\) solve the system by the method of elimination. Label each line with its equation. To print an enlarged copy of the graph, go to Math Graphs.com. $$ \left\\{\begin{array}{c}{x+y=0} \\ {3 x+2 y=1}\end{array}\right. $$

Solving a System by Substitution In Exercises \(7-14,\) solve the system by the method of substitution. Check your solution(s) graphically. $$\left\\{\begin{array}{l}{2 x+y=6} \\ {-x+y=0}\end{array}\right.$$

Matching. Match the rational expression with the form of its decomposition. IThe decompositions are labeled (a), (b), \((\mathrm{c}),\) and (d).] \(\begin{array}{ll}{\text { (a) } \frac{A}{x}+\frac{B}{x+2}+\frac{C}{x-2}} & {\text { (b) } \frac{A}{x}+\frac{B}{x-4}} \\ {\text { (c) } \frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x-4}} & {\text { (d) } \frac{A}{x}+\frac{B x+C}{x^{2}+4}}\end{array}\) $$\frac{3 x-1}{x\left(x^{2}+4\right)}$$

Checking Solutions In Exercises \(7 - 10\) , determine whether each ordered triple is a solution of the system of equations. $$\left\\{ \begin{aligned} 6 x - y + z & = - 1 \\ 4 x - 3 z & = - 19 \\ 2 y + 5 z & = 25 \end{aligned} \right.$$ $$\begin{array} { l l } { \text { (a) } ( 2,0 , - 2 ) } & { \text { (b) } ( - 3,0,5 ) } \\ { \text { (c) } ( 0 , - 1,4 ) } & { \text { (d) } ( - 1,0,5 ) } \end{array}$$

Solving a Linear Programming Problem, find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) $$ \begin{array}{c}{\text { Objective function: }} \\ {z=2 x+8 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {2 x+y \leq 4}\end{array} $$

In Exercises 5-18, sketch the graph of the inequality. $$x<-4$$

Solving a System by Elimination In Exercises \(5-12,\) solve the system by the method of elimination. Label each line with its equation. To print an enlarged copy of the graph, go to MathGraphs.com. $$ \left\\{\begin{array}{l}{2 x-y=-3} \\ {4 x+3 y=-21}\end{array}\right. $$

Solving a System by Substitution In Exercises \(7-14,\) solve the system by the method of substitution. Check your solution(s) graphically. $$\left\\{\begin{array}{rr}{x-4 y} & {=-11} \\ {x+3 y} & {=\quad 3}\end{array}\right.$$

Matching. Match the rational expression with the form of its decomposition. IThe decompositions are labeled (a), (b), \((\mathrm{c}),\) and (d).] \(\begin{array}{ll}{\text { (a) } \frac{A}{x}+\frac{B}{x+2}+\frac{C}{x-2}} & {\text { (b) } \frac{A}{x}+\frac{B}{x-4}} \\ {\text { (c) } \frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x-4}} & {\text { (d) } \frac{A}{x}+\frac{B x+C}{x^{2}+4}}\end{array}\) $$\frac{3 x-1}{x\left(x^{2}-4\right)}$$

Checking Solutions In Exercises \(7 - 10\) , determine whether each ordered triple is a solution of the system of equations. $$\left\\{ \begin{aligned} 3 x + 4 y - z = & 17 \\ 5 x - y + 2 z = & \- 2 \\\ 2 x - 3 y + 7 z = & \- 21 \end{aligned} \right.$$ $$\begin{array} { l l } { \text { (a) } ( 3 , - 1,2 ) } & { \text { (b) } ( 1,3 , - 2 ) } \\ { \text { (c) } ( 4,1 , - 3 ) } & { \text { (d) } ( 1 , - 2,2 ) } \end{array}$$

Solving a Linear Programming Problem, find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) $$ \begin{array}{c}{\text { Objective function: }} \\ {z=2 x+5 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {x+3 y \leq 15} \\ {4 x+y \leq 16}\end{array} $$

In Exercises 5-18, sketch the graph of the inequality. $$y>-7$$

Solving a System by Elimination In Exercises \(5-12,\) solve the system by the method of elimination. Label each line with its equation. To print an enlarged copy of the graph, go to Math Graphs.com. $$ \left\\{\begin{array}{c}{x-y=2} \\ {-2 x+2 y=5}\end{array}\right. $$

Solving a System by Substitution In Exercises \(7-14,\) solve the system by the method of substitution. Check your solution(s) graphically. $$\left\\{\begin{array}{c}{x-y=-4} \\ {x^{2}-y=-2}\end{array}\right.$$

Writing the Form of the Decomposition. Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{3}{x^{2}-2 x}$$

Checking Solutions In Exercises \(7 - 10\) , determine whether each ordered triple is a solution of the system of equations. $$\left\\{ \begin{aligned} 4 x + y - z & = 0 \\ - 8 x - 6 y + z & = - \frac { 2 } { 4 } \\ 3 x - y & = - \frac { 9 } { 4 } \end{aligned} \right.$$ $$\begin{array} { l l } { \text { (a) } \left( \frac { 1 } { 2 } , - \frac { 3 } { 4 } , - \frac { 7 } { 4 } \right) } & { \text { (b) } \left( - \frac { 3 } { 2 } , \frac { 5 } { 4 } , - \frac { 5 } { 4 } \right) } \\ { \text { (c) } \left( - \frac { 1 } { 2 } , \frac { 3 } { 4 } , - \frac { 5 } { 4 } \right) } & { \text { (d) } \left( - \frac { 1 } { 2 } , \frac { 1 } { 6 } , - \frac { 3 } { 4 } \right) } \end{array}$$

Solving a Linear Programming Problem, find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) $$ \begin{array}{c}{\text { Objective function: }} \\ {z=4 x+5 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {2 x+3 y \geq 6} \\ {3 x+y \leq 9} \\\ {x+4 y \leq 16}\end{array} $$

In Exercises 5-18, sketch the graph of the inequality. $$10 \geq y$$

Solving a System by Elimination In Exercises \(5-12,\) solve the system by the method of elimination. Label each line with its equation. To print an enlarged copy of the graph, go to Math Graphs.com. $$ \left\\{\begin{array}{l}{3 x+2 y=3} \\ {6 x+4 y=14}\end{array}\right. $$