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Chapter 8: Problem 46

Let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The difference between the squares of two numbers is \(5 .\) Twice the square of the second number subtracted from three times the square of the first number is \(19 .\) Find the numbers.

Short Answer

Expert verified
The pairs of numbers that satisfy the given conditions are \((x, y)= (3,2), (-3,-2), (-3, 2), (3,-2)\).

Step by step solution

01

Define parameters and write the equations

From the exercise, we get two equations. 1). The difference between the squares of two numbers is \(5\). So, \(x^{2}-y^{2}=5\). 2). Twice the square of the second number subtracted from three times the square of the first number is \(19\). This translates to \(3x^{2}-2y^{2}=19\). The system of equations then becomes: \[\begin{cases}x^{2} - y^{2} = 5 \\3x^{2} - 2y^{2} = 19\end{cases}\]
02

Isolate one of the variables

Let's isolate \(x^{2}\) in the first equation, \(x^{2}=y^{2}+5\).
03

Substitute in the second equation

We substitute our isolated variable \(x^{2}\) from the first equation into the second equation. This gives us: \[3(y^{2}+5) - 2y^{2} = 19\].
04

Solve the equation for y

Solving the equation for \(y\), we simplify the equation to: \(y^{2}+15-19=0\), which can be rewritten as \(y^{2}-4=0\). Solving for \(y\) we obtain two solutions: \(y=2, -2\).
05

Substitute the y values in the first equation

Substitute each \(y\) value into \(x^{2}=y^{2}+5\) to solve for \(x\). For \(y=2\), we get \(x^{2}=2^{2}+5=9\), so \(x=3, -3\). In the other case, for \(y=-2\), we get \(x^{2}=(-2)^{2}+5=9\), so \(x=3, -3\). Therefore, the pairs of solutions to the problem are \((x, y)= (3,2), (-3,-2), (-3, 2), (3,-2)\).
06

Verify the solution

The solutions can be verified by substituting them back into the original equations to ensure they satisfy both conditions.

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Most popular questions from this chapter

a. A student earns \(\$ 15\) per hour for tutoring and \(\$ 10\) per hour as a teacher's aide. Let \(x=\) the number of hours each week spent tutoring and let \(y=\) the number of hours each week spent as a teacher's aide. Write the objective function that models total weekly earnings. b. The student is bound by the following constraints: \(\cdot\) To have enough time for studies, the student can work no more than 20 hours per week. \(\cdot\) The tutoring center requires that each tutor spend at least three hours per week tutoring. \(\cdot\) The tutoring center requires that each tutor spend no more than eight hours per week tutoring. Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) are nonnegative. d. Evaluate the objective function for total weekly earnings at each of the four vertices of the graphed region. [The vertices should occur at \((3,0),(8,0),(3,17),\) and \((8,12) .]\) e. Complete the missing portions of this statement: The student can earn the maximum amount per week by tutoring for ____ hours per week and working as a teacher's aide for ____ hours per week. The maximum amount that the student can earn each week is $_____.

Expand: \(\log _{8}\left(\frac{\sqrt[4]{x}}{64 y^{3}}\right) .\) (Section 4.3, Example 4)

At the north campus of a performing arts school, 10% of the students are music majors. At the south campus, 90% of the students are music majors. The campuses are merged into one east campus. If 42% of the 1000 students at the east campus are music majors, how many students did each of the north and south campuses have before the merger?

Will help you prepare for the material covered in the next section. a. Graph the solution set of the system: \(\left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x-2 y & \leq 6 \\ y & \leq-x+7 \end{aligned}\right.\) b. List the points that form the corners of the graphed region in part (a). c. Evaluate \(2 x+5 y\) at each of the points obtained in part (b).

Solve: \(\sqrt{2 x-5}-\sqrt{x-3}=1 .\) (Section \(1.6,\) Example 4 )
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