Chapter 5

Solve the system by elimination. Then use the graph to confirm your solution. Copy the graph and label each line with the appropriate equation. $$ \left\\{\begin{aligned} 3 x-2 y &=2 \\ x+2 y &=6 \end{aligned}\right. $$

Determine whether each ordered pair is a solution of the system of equations. \(\left\\{\begin{array}{rr}x+4 y= & -3 \\ 5 x-y= & 6\end{array}\right.\) (a) \((-1,-1)\) (b) \((1,-1)\)

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.) Objective function: $$ z=2 x+8 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 2 x+y & \leq 4 \end{aligned} $$

Determine whether each ordered pair is a solution of the system of equations. \(\left\\{\begin{aligned} 2 x-y &=2 \\ x+3 y &=8 \end{aligned}\right.\) (a) \((2,1)\) (b) \((2,2)\)

Determine whether each ordered pair is a solution of the system of equations. \(\left\\{\begin{array}{l}2 x+5 y=-5 \\ 2 x-y^{2}=1\end{array}\right.\) (a) \((5,-3)\) (b) \((0,-1)\)

Match each system of equations with its solution. [The solutions are labeled (a), (b), (c), and (d).] (a) \((-1,0,3)\) (b) \((6,2,-2)\) (c) \((2,1,-3)\) (d) \((4,-1,5)\) $$ \left\\{\begin{aligned} -x-2 y+5 z &=23 \\ -3 x+y+6 z &=17 \\ 9 x+2 y-7 z &=-1 \end{aligned}\right. $$

Determine whether each ordered pair is a solution of the system of equations. \(\left\\{\begin{array}{l}4 x^{2}+y=3 \\ -x-y=11\end{array}\right.\) (a) \((-2,-9)\) (b) \((2,-13)\)

Determine whether the system of equations is in row-echelon form. Justify your answer. $$ \left\\{\begin{array}{rr} x+3 y-7 z= & -11 \\ y-2 z= & -3 \\ z= & 2 \end{array}\right. $$

Determine whether each ordered pair is a solution of the system of equations. \(\left\\{\begin{array}{l}y=-2 e^{x} \\ 3 x-y=2\end{array}\right.\) (a) \((-2,0)\) (b) \((-1,2)\)

Determine whether the system of equations is in row-echelon form. Justify your answer. $$ \left\\{\begin{array}{r} x-y+3 z=-11 \\ y+8 z=-12 \\ z=-2 \end{array}\right. $$

Sketch the graph of the inequality. $$x \geq 2$$

Determine whether the system of equations is in row-echelon form. Justify your answer. $$ \left\\{\begin{aligned} x-9 y+z &=22 \\ 2 y+z &=-3 \\ z &=1 \end{aligned}\right. $$

Sketch the graph of the inequality. $$x<4$$

Determine whether the system of equations is in row-echelon form. Justify your answer. $$ \left\\{\begin{array}{rr} x-y-8 z= & 12 \\ 2 y-2 z= & 2 \\ 7 z= & -7 \end{array}\right. $$

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $$ z=6 x+10 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x+5 y & \leq 15 \end{aligned} $$

Sketch the graph of the inequality. $$y+2 x^{2}>0$$

Use back-substitution to solve the system of linear equations. $$\left\\{\begin{aligned} x-y+z &=4 \\ 2 y+z &=-6 \\ z &=-2 \end{aligned}\right.$$

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $$ z=7 x+8 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+2 y & \leq 8 \end{aligned} $$

Sketch the graph of the inequality. $$y^{2}-x<0$$

Use back-substitution to solve the system of linear equations. $$\left\\{\begin{aligned} 4 x-2 y+z &=8 \\\\-y+z &=4 \\ z &=2 \end{aligned}\right.$$

Sketch the graph of the inequality. $$y>-1$$

Solve the system of equations. $$\left\\{\begin{aligned} 4 x+y-3 z &=11 \\ 2 x-3 y+2 z &=9 \\ x+y+z &=-3 \end{aligned}\right.$$

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}x+2 y=3 \\ x-2 y=1\end{array}\right.$$

Sketch the graph of the inequality. $$y \leq 3$$

Solve the system of equations. $$\left\\{\begin{aligned} 6 y+4 z &=-12 \\ 3 x+3 y &=9 \\ 2 x-3 z &=10 \end{aligned}\right.$$

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{aligned} 2 x-3 y &=4 \\\\-2 x-y &=4 \end{aligned}\right.$$

Objective function: $$ z=4 x+5 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+y & \geq 8 \\ 3 x+5 y & \geq 30 \end{aligned} $$

Sketch the graph of the inequality. $$y<2-x$$

Solve the system of equations. $$\left\\{\begin{aligned} 3 x+2 z &=13 \\ x+2 y+z &=-5 \\\\-3 y-z &=10 \end{aligned}\right.$$

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{aligned} 4 x-3 y &=11 \\\\-6 x+3 y &=3 \end{aligned}\right.$$

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: \(z=4 x+5 y\) Constraints: \(\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+y & \leq 5 \\ x+2 y & \leq 6 \end{aligned}\)

Sketch the graph of the inequality. $$y>2 x-4$$

Solve the system of equations. $$\left\\{\begin{array}{rr}2 x+3 y+z= & -4 \\ 2 x-4 y+3 z= & 18 \\ 3 x-2 y+2 z= & 9\end{array}\right.$$

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}3 x-5 y=2 \\ 2 x+5 y=13\end{array}\right.$$

Sketch the graph of the inequality. $$2 y-x \geq 4$$

Solve the system of equations. $$\left\\{\begin{aligned} 3 x-2 y+4 z &=1 \\ x+y-2 z &=3 \\ 2 x-3 y+6 z &=8 \end{aligned}\right.$$

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}3 x-y=17 \\ 5 x+5 y=-5\end{array}\right.$$

Solve the system by the method of substitution. Then use the graph to confirm your solution. $$ \left\\{\begin{array}{l} y=-x^{2}+1 \\ y=x^{2}-1 \end{array}\right. $$

Sketch the graph of the inequality. $$5 x+3 y \geq-15$$

Solve the system of equations. $$\left\\{\begin{aligned} 5 x-3 y+2 z &=3 \\ 2 x+4 y-z &=7 \\ x-11 y+4 z &=3 \end{aligned}\right.$$

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{r}x+7 y=12 \\ 3 x-5 y=10\end{array}\right.$$

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $$ z=x+2 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+2 y & \leq 40 \\ x+y & \leq 30 \\ 2 x+3 y & \leq 65 \end{aligned} $$

Sketch the graph of the inequality. $$y<\ln x$$

Solve the system of equations. $$\left\\{\begin{array}{l}3 x+3 y+5 z=1 \\ 3 x+5 y+9 z=0 \\ 5 x+9 y+17 z=0\end{array}\right.$$

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}3 x+2 y=10 \\ 2 x+5 y=3\end{array}\right.$$

Solve the system by the method of substitution. $$\left\\{\begin{aligned} 2 x-y &=-3 \\\\-3 x-4 y &=-1 \end{aligned}\right.$$

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $$ z=x $$ Constraints: $$ \begin{array}{r} x \geq 0 \\ y \geq 0 \\ 2 x+3 y \leq 60 \\ 2 x+y \leq 28 \\ 4 x+y \leq 48 \end{array} $$

Sketch the graph of the inequality. $$y \geq-\ln x+1$$

Solve the system of equations. $$\left\\{\begin{aligned} 2 x+y-z &=13 \\ x+2 y+z &=2 \\ 8 x-3 y+4 z &=-2 \end{aligned}\right.$$

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{r}8 r+16 s=20 \\ 16 r+50 s=55\end{array}\right.$$