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Chapter 7: Problem 37

In Exercises 37-44, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(\left\\{\begin{array}{l}x=9-2 y \\ x+2 y=13\end{array}\right.\)

Short Answer

Expert verified
The solution to the system of equations is \(\emptyset\) (no solution)

Step by step solution

01

Expression Simplification

Simplify the second equation of the system, \(x+2 y=13\), to match the form of the first equation, \(x=9-2y\). This can be done by subtracting \(2y\) from both sides. You should obtain \(x=13-2y\)
02

Equate the Expressions

The expressions for both \(x\) obtained from the two equations can be set equal to each other, thereby equating \(9-2y=13-2y\)
03

Solve for y

Try to solve for \(y\) by subtracting \(9\) from both sides. However, as the \(2y\) terms cancel each other out, you're left with \(0 = 4\), which is a contradiction. The contradiction implies that there is no solution for this system of equations.
04

Express the Solution in Set Notation

Express the solutions in set notation, which in this case is an empty set as there are no solutions. That is, \(\emptyset\)

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Most popular questions from this chapter

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, \(y\), in feet, in terms of its horizontal distance, \(x\), in feet. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \boldsymbol{x}, \text { Ball's } \\ \text { Horizontal } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \boldsymbol{y}, \\ \text { Ball's } \\ \text { Height } \\ \text { (feet) } \end{array} \\ \hline 0 & 6 \\ \hline 0.5 & 7.4 \\ \hline 1.5 & 9 \\ \hline 4 & 6 \\ \hline \end{array} $$ $$ \begin{array}{|l|} \hline u a d R e g \\ y=a \times 2+b x+c \\ a=-.8 \\ b=3.2 \\ c=6 \end{array} $$ a. Explain why a quadratic function was used to model the data. Why is the value of \(a\) negative? b. Use the graphing calculator screen to express the model in function notation. c. Use the model from part (b) to determine the \(x\)-coordinate of the quadratic function's vertex. Then complete this statement: The maximum height of the ball occurs _____ feet from where it was thrown, and the maximum height is _____ feet.

Graph each linear inequality. \(3 x+y \leq 3\)

The figure shows the healthy weight region for various heights for people ages 35 and older. If \(x\) represents height, in inches, and y represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: $$ \left\\{\begin{array}{l} 5.3 x-y \geq 180 \\ 4.1 x-y \leq 140 \end{array}\right. $$ Use this information to solve Exercises 45-48. Is a person in this age group who is 5 feet 8 inches tall weighing 135 pounds within the healthy weight region?

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & -3 \\ \hline 1 & 2 \\ \hline 2 & 7 \\ \hline 3 & 12 \\ \hline 4 & 17 \\ \hline \end{array} $$

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 4 \\ \hline 1 & 5 \\ \hline 2 & 7 \\ \hline 3 & 11 \\ \hline 4 & 19 \\ \hline \end{array} $$
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