Find the values of $x \geq 0$ and $y \geq 0$ that maximize $z=14 x+13 y$ subject to each of the following sets of constraints. $\newline$ (a) $\newline$ $\begin{array}{l} x+y \leq 19 \\ x+2 y \leq 22 \end{array}$ $\newline$ (b) $3 x+y \leq 12$ $\newline$ $\begin{array}{l} \text { (c) } 2 x+5 y \geq 18 \\ 3 x+2 y \leq 15 \\ 2 \mathrm{x}+2 \mathrm{y} \leq 13 \\ \end{array}$ $\newline$ $x+2 y \leq 20$ $\newline$ (a) Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. $\newline$ A. The maximum value is $\square$ and occurs at the point $\square$ (Simplify your answers.) $\newline$ B. There is no maximum value.

View step-by-step help

Home

Math Problems

Grade 7

One-step inequalities: word problems

Full solution

Q. Find the values of

x \geq 0

and

y \geq 0

that maximize

z=14 x+13 y

subject to each of the following sets of constraints.

\newline

(a)

\newline

\begin{array}{l} x+y \leq 19 \\ x+2 y \leq 22 \end{array}

\newline

(b)

3 x+y \leq 12

\newline

\begin{array}{l} \text { (c) } 2 x+5 y \geq 18 \\ 3 x+2 y \leq 15 \\ 2 \mathrm{x}+2 \mathrm{y} \leq 13 \\ \end{array}

\newline

x+2 y \leq 20

\newline

(a) Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

\newline

A. The maximum value is

\square

and occurs at the point

\square

(Simplify your answers.)

\newline

B. There is no maximum value.

Intersection Points Calculation: For part (a), we need to find the intersection points of the lines $x+y=19$ and $x+2y=22$ . These points, along with the axes intercepts, will be our candidates for the maximum value of $z$ .
Finding Intercepts: First, let's find the $x$ -intercept for $x+y=19$ by setting $y=0$ , which gives us $x=19$ .
Solving System of Equations: Now, let's find the y-intercept for $x+y=19$ by setting $x=0$ , which gives us $y=19$ .
Testing Points in Objective Function: Next, we find the $x$ -intercept for $x+2y=22$ by setting $y=0$ , which gives us $x=22$ .
Maximum Value Determination: Then, we find the y-intercept for $x+2y=22$ by setting $x=0$ , which gives us $y=11$ .
Finding Intercepts: Now we solve the system of equations to find the intersection point of $x+y=19$ and $x+2y=22$ . Subtracting the first equation from the second gives us $y=3$ .
Testing Points in Objective Function: Substitute $y=3$ into $x+y=19$ to find $x$ , which gives us $x=16$ .
Maximum Value Determination: We now have the points $19,0$ , $0,19$ , $22,0$ , $0,11$ , and $16,3$ to test in the objective function $z=14x+13y$ .
Feasible Region Analysis: Plugging $(19,0)$ into $z$ gives us $z=14(19)+13(0)=266$ .
Finding Intersection Points: Plugging $(0,19)$ into $z$ gives us $z=14(0)+13(19)=247$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ . The maximum value from these points is $308$ , which occurs at $(22,0)$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ . The maximum value from these points is $308$ , which occurs at $(22,0)$ . For part (b), we find the $z$ $1$ -intercept of $z$ $2$ by setting $z$ $3$ , which gives us $z$ $4$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ . The maximum value from these points is $308$ , which occurs at $(22,0)$ . For part (b), we find the $z$ $1$ -intercept of $z$ $2$ by setting $z$ $3$ , which gives us $z$ $4$ . Now, we find the $z$ $5$ -intercept of $z$ $2$ by setting $z$ $7$ , which gives us $z$ $8$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ . The maximum value from these points is $308$ , which occurs at $(22,0)$ . For part (b), we find the $z$ $1$ -intercept of $z$ $2$ by setting $z$ $3$ , which gives us $z$ $4$ . Now, we find the $z$ $5$ -intercept of $z$ $2$ by setting $z$ $7$ , which gives us $z$ $8$ . We test the points $z$ $9$ and $z=14(22)+13(0)=308$ $0$ in the objective function $z=14(22)+13(0)=308$ $1$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ . The maximum value from these points is $308$ , which occurs at $(22,0)$ . For part (b), we find the $z$ $1$ -intercept of $z$ $2$ by setting $z$ $3$ , which gives us $z$ $4$ . Now, we find the $z$ $5$ -intercept of $z$ $2$ by setting $z$ $7$ , which gives us $z$ $8$ . We test the points $z$ $9$ and $z=14(22)+13(0)=308$ $0$ in the objective function $z=14(22)+13(0)=308$ $1$ . Plugging $z$ $9$ into $z$ gives us $z=14(22)+13(0)=308$ $4$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ . The maximum value from these points is $308$ , which occurs at $(22,0)$ . For part (b), we find the $z$ $1$ -intercept of $z$ $2$ by setting $z$ $3$ , which gives us $z$ $4$ . Now, we find the $z$ $5$ -intercept of $z$ $2$ by setting $z$ $7$ , which gives us $z$ $8$ . We test the points $z$ $9$ and $z=14(22)+13(0)=308$ $0$ in the objective function $z=14(22)+13(0)=308$ $1$ . Plugging $z$ $9$ into $z$ gives us $z=14(22)+13(0)=308$ $4$ . Plugging $z=14(22)+13(0)=308$ $0$ into $z$ gives us $z=14(22)+13(0)=308$ $7$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ . The maximum value from these points is $308$ , which occurs at $(22,0)$ . For part (b), we find the $z$ $1$ -intercept of $z$ $2$ by setting $z$ $3$ , which gives us $z$ $4$ . Now, we find the $z$ $5$ -intercept of $z$ $2$ by setting $z$ $7$ , which gives us $z$ $8$ . We test the points $z$ $9$ and $z=14(22)+13(0)=308$ $0$ in the objective function $z=14(22)+13(0)=308$ $1$ . Plugging $z$ $9$ into $z$ gives us $z=14(22)+13(0)=308$ $4$ . Plugging $z=14(22)+13(0)=308$ $0$ into $z$ gives us $z=14(22)+13(0)=308$ $7$ . The maximum value from these points is $z=14(22)+13(0)=308$ $8$ , which occurs at $z=14(22)+13(0)=308$ $0$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ . The maximum value from these points is $308$ , which occurs at $(22,0)$ . For part (b), we find the $z$ $1$ -intercept of $z$ $2$ by setting $z$ $3$ , which gives us $z$ $4$ . Now, we find the $z$ $5$ -intercept of $z$ $2$ by setting $z$ $7$ , which gives us $z$ $8$ . We test the points $z$ $9$ and $z=14(22)+13(0)=308$ $0$ in the objective function $z=14(22)+13(0)=308$ $1$ . Plugging $z$ $9$ into $z$ gives us $z=14(22)+13(0)=308$ $4$ . Plugging $z=14(22)+13(0)=308$ $0$ into $z$ gives us $z=14(22)+13(0)=308$ $7$ . The maximum value from these points is $z=14(22)+13(0)=308$ $8$ , which occurs at $z=14(22)+13(0)=308$ $0$ . For part (c), we have a system of inequalities. We need to find the feasible region and test the corner points in the objective function $z=14(22)+13(0)=308$ $1$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ . The maximum value from these points is $308$ , which occurs at $(22,0)$ . For part (b), we find the $z$ $1$ -intercept of $z$ $2$ by setting $z$ $3$ , which gives us $z$ $4$ . Now, we find the $z$ $5$ -intercept of $z$ $2$ by setting $z$ $7$ , which gives us $z$ $8$ . We test the points $z$ $9$ and $z=14(22)+13(0)=308$ $0$ in the objective function $z=14(22)+13(0)=308$ $1$ . Plugging $z$ $9$ into $z$ gives us $z=14(22)+13(0)=308$ $4$ . Plugging $z=14(22)+13(0)=308$ $0$ into $z$ gives us $z=14(22)+13(0)=308$ $7$ . The maximum value from these points is $z=14(22)+13(0)=308$ $8$ , which occurs at $z=14(22)+13(0)=308$ $0$ . For part (c), we have a system of inequalities. We need to find the feasible region and test the corner points in the objective function $z=14(22)+13(0)=308$ $1$ . First, we find the intersection points of the lines $(0,11)$ $1$ , $(0,11)$ $2$ , and $(0,11)$ $3$ with each other and with the axes.
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ . The maximum value from these points is $308$ , which occurs at $(22,0)$ . For part (b), we find the $z$ $1$ -intercept of $z$ $2$ by setting $z$ $3$ , which gives us $z$ $4$ . Now, we find the $z$ $5$ -intercept of $z$ $2$ by setting $z$ $7$ , which gives us $z$ $8$ . We test the points $z$ $9$ and $z=14(22)+13(0)=308$ $0$ in the objective function $z=14(22)+13(0)=308$ $1$ . Plugging $z$ $9$ into $z$ gives us $z=14(22)+13(0)=308$ $4$ . Plugging $z=14(22)+13(0)=308$ $0$ into $z$ gives us $z=14(22)+13(0)=308$ $7$ . The maximum value from these points is $z=14(22)+13(0)=308$ $8$ , which occurs at $z=14(22)+13(0)=308$ $0$ . For part (c), we have a system of inequalities. We need to find the feasible region and test the corner points in the objective function $z=14(22)+13(0)=308$ $1$ . First, we find the intersection points of the lines $(0,11)$ $1$ , $(0,11)$ $2$ , and $(0,11)$ $3$ with each other and with the axes. The $z$ $1$ -intercept for $(0,11)$ $1$ is found by setting $z$ $3$ , which gives us $(0,11)$ $7$ .
Intercepts Calculation: Plugging $(22,0)$ into $z$ gives us $z=14(22)+13(0)=308$ . Plugging $(0,11)$ into $z$ gives us $z=14(0)+13(11)=143$ . Plugging $(16,3)$ into $z$ gives us $z=14(16)+13(3)=287$ . The maximum value from these points is $308$ , which occurs at $(22,0)$ . For part (b), we find the x-intercept of $z$ $1$ by setting $z$ $2$ , which gives us $z$ $3$ . Now, we find the y-intercept of $z$ $1$ by setting $z$ $5$ , which gives us $z$ $6$ . We test the points $z$ $7$ and $z$ $8$ in the objective function $z$ $9$ . Plugging $z$ $7$ into $z$ gives us $z=14(22)+13(0)=308$ $2$ . Plugging $z$ $8$ into $z$ gives us $z=14(22)+13(0)=308$ $5$ . The maximum value from these points is $z=14(22)+13(0)=308$ $6$ , which occurs at $z$ $8$ . For part (c), we have a system of inequalities. We need to find the feasible region and test the corner points in the objective function $z$ $9$ . First, we find the intersection points of the lines $z=14(22)+13(0)=308$ $9$ , $(0,11)$ $0$ , and $(0,11)$ $1$ with each other and with the axes. The x-intercept for $z=14(22)+13(0)=308$ $9$ is found by setting $z$ $2$ , which gives us $(0,11)$ $4$ . The y-intercept for $z=14(22)+13(0)=308$ $9$ is found by setting $z$ $5$ , which gives us $(0,11)$ $7$ , but since $(0,11)$ $8$ must be non-negative integer, we round down to $(0,11)$ $9$ .