Chapter 5

In Exercises \(1-4,\) determine whether the given ordered pair is a solution of the system. $$ \begin{aligned} &(2,3)\\\ &x+3 y=11\\\ &x-5 y=-13 \end{aligned} $$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{11 x-10}{(x-2)(x+1)}$$

Graph each inequality. $$ x+2 y \leq 8 $$

Solve each system by the substitution method. $$\begin{aligned} &x+y=2\\\ &y=x^{2}-4 \end{aligned}$$

Determine if the given ordered triple is a solution of the system. $$\begin{array}{rr}x+y+z= & 4 \\\x-2 y-z= & 1 \\\2 x-y-2 z= & -1\\\\(2,-1,3) &\end{array}$$

In Exercises \(1-4,\) determine whether the given ordered pair is a solution of the system. $$ \begin{aligned} &(-3,5)\\\ &9 x+7 y=8\\\ &8 x-9 y=-69 \end{aligned} $$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{5 x+7}{(x-1)(x+3)}$$

Graph each inequality. $$ 3 x-6 y \leq 12 $$

Solve each system by the substitution method. $$\begin{aligned} &x-y=-1\\\ &y=x^{2}+1 \end{aligned}$$

Determine if the given ordered triple is a solution of the system. $$\begin{array}{rr}x+y+z= & 0 \\\x+2 y-3 z= & 5 \\\3 x+4 y+2 z= & -1 \\\\(5,-3,-2) &\end{array}$$

In Exercises \(1-4,\) determine whether the given ordered pair is a solution of the system. $$ \begin{aligned} (2,5) & \\ 2 x+3 y &=17 \\ x+4 y &=16 \end{aligned} $$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{6 x^{2}-14 x-27}{(x+2)(x-3)^{2}}$$

Graph each inequality. $$ x-2 y>10 $$

Determine if the given ordered triple is a solution of the system. $$\begin{aligned}x-2y\quad &=2 \\\2 x+3 y\quad &=11 \\\y-4 z &=-7 \\\\(4,1,2) &\end{aligned}$$

Solve each system by the substitution method. $$\begin{aligned} &x+y=2\\\ &y=x^{2}-4 x+4 \end{aligned}$$

In Exercises \(1-4,\) determine whether the given ordered pair is a solution of the system. $$ \begin{aligned} &(8,5)\\\ &5 x-4 y=20\\\ &3 y=2 x+1 \end{aligned} $$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{3 x+16}{(x+1)(x-2)^{2}}$$

Graph each inequality. $$ 2 x-y>4 $$

Determine if the given ordered triple is a solution of the system. $$\begin{aligned}&x-2 z=-5\\\&y-3 z=-3\\\&2 x-z=-4\\\&(-1,3,2)\end{aligned}$$

Solve each system by the substitution method. $$\begin{aligned} &2 x+y=-5\\\ &y=x^{2}+6 x+7 \end{aligned}$$

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x+y=4\\\ &y=3 x \end{aligned} $$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{5 x^{2}-6 x+7}{(x-1)\left(x^{2}+1\right)}$$

Graph each inequality. $$ y \leq \frac{1}{3} x $$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function \(z=3 x+2 y\) Constraints \(\quad x \geq 0, y \geq 0\) \(2 x+y \leq 8\) \(x+y \geq 4\)

Solve each system. $$\begin{aligned}&x+y+2 z=11\\\&x+y+3 z=14\\\&x+2 y-z=5\end{aligned}$$

Solve each system by the substitution method. $$\begin{aligned} &y=x^{2}-4 x-10\\\ &y=-x^{2}-2 x+14 \end{aligned}$$

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x+y=6\\\ &y=2 x \end{aligned} $$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{5 x^{2}-9 x+19}{(x-4)\left(x^{2}+5\right)}$$

Graph each inequality. $$ y \leq \frac{1}{4} x $$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function \(\quad z=2 x+3 y\) \(\begin{aligned} \text { Constraints } & & x \geq 0, y & \geq 0 \\ & & 2 x+y & \leq 8 \\ & & & 2 x+3 y & \leq 12 \end{aligned}\)

Solve each system. $$\begin{aligned}2 x+y-2 z &=-1 \\\3 x-3 y-z &=5 \\\x-2 y+3 z &=6\end{aligned}$$

Solve each system by the substitution method. $$\begin{aligned} &y=x^{2}+4 x+5\\\ &y=x^{2}+2 x-1 \end{aligned}$$

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x+3 y=8\\\ &y=2 x-9 \end{aligned} $$

Graph each inequality. $$ y>2 x-1 $$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function \(\quad z=4 x+y\) Constraints \(\quad x \geq 0, y \geq 0\) \(2 x+3 y \leq 12\) \(x+y \geq 3\)

Solve each system. $$\begin{array}{r}4 x-y+2 z=11 \\\x+2 y-z=-1 \\\2 x+2 y-3 z=-1\end{array}$$

Solve each system by the substitution method. $$\begin{aligned} &x^{2}+y^{2}=25\\\ &x-y=1 \end{aligned}$$

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &2 x-3 y=-13\\\ &y=2 x+7 \end{aligned} $$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{7 x^{2}-9 x+3}{\left(x^{2}+7\right)^{2}}$$

Graph each inequality. $$ y>3 x+2 $$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function \(\quad z=x+6 y\) Constraints \(\quad x \geq 0, y \geq 0\) $$\begin{array}{l} 2 x+y \leq 10 \\ x-2 y \geq-10 \end{array}$$

Solve each system. $$\begin{aligned}x-y+3 z &=8 \\\3 x+y-2 z &=-2 \\\2 x+4 y+z &=0\end{aligned}$$

Solve each system by the substitution method. $$\begin{aligned} &x^{2}+y^{2}=5\\\ &3 x-y=5 \end{aligned}$$

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{array}{l} x=4 y-2 \\ x=6 y+8 \end{array} $$

Graph each inequality. $$ x \leq 1 $$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function \(\quad z=3 x-2 y\) Constraints \(\quad 1 \leq x \leq 5\) \(y \geq 2\) \(x-y \geq-3\)

Solve each system. $$\begin{aligned}3 x+5 y+2 z &=0 \\\12 x-15 y+4 z &=12 \\\6 x-25 y-8 z &=8\end{aligned}$$

Solve each system by the substitution method. $$\begin{array}{l} x y=6 \\ 2 x-y=1 \end{array}$$

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &x=3 y+7\\\ &x=2 y-1 \end{aligned} $$

Graph each inequality. $$ x \leq-3 $$