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Chapter 9: Problem 32

Give all solutions of each nonlinear system of equations, including those with nonreal complex components. $$\begin{aligned} &-5 x y+2=0\\\ &x-15 y=5 \end{aligned}$$

Short Answer

Expert verified
The solutions are \((6, \frac{1}{15})\) and \((-1, -\frac{2}{5})\).

Step by step solution

01

Solve the second equation for x

Given the second equation: \[ x - 15y = 5 \] Solve for \(x\). \[ x = 5 + 15y \]
02

Substitute x into the first equation

Substitute the expression for \(x\) from Step 1 into the first equation: \[ -5xy + 2 = 0 \] Replace \(x\): \[ -5(5 + 15y)y + 2 = 0 \]
03

Simplify and solve for y

Simplify the equation: \[ -5(5y + 15y^2) + 2 = 0 \] Distribute and combine like terms: \[ -25y - 75y^2 + 2 = 0 \] Rearrange: \[ -75y^2 - 25y + 2 = 0 \] Divide the entire equation by -1: \[ 75y^2 + 25y - 2 = 0 \] Solve this quadratic equation using the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 75\), \(b = 25\), and \(c = -2\): \[ y = \frac{-25 \pm \sqrt{625 + 600}}{150} \] \[ y = \frac{-25 \pm \sqrt{1225}}{150} \] \[ y = \frac{-25 \pm 35}{150} \] This gives two solutions: \[ y = \frac{10}{150} = \frac{1}{15} \] \[ y = \frac{-60}{150} = -\frac{2}{5} \]
04

Find corresponding x values

Using the solutions for \(y\), find corresponding \(x\) values: For \(y = \frac{1}{15}\): \[ x = 5 + 15\left(\frac{1}{15}\right) = 6 \] For \(y = -\frac{2}{5}\): \[ x = 5 + 15\left(-\frac{2}{5}\right) = -1 \]
05

Verify the solutions in the original equations

Substitute \((6, \frac{1}{15})\) and \((-1, -\frac{2}{5})\) back into the original equations to verify they satisfy both: For \((6, \frac{1}{15})\): \[ -5(6)(\frac{1}{15}) + 2 = 0 \] \[ 6 - 15(\frac{1}{15}) = 5 \] For \((-1, -\frac{2}{5})\): \[ -5(-1)(-\frac{2}{5}) + 2 = 0 \] \[ -1 - 15(-\frac{2}{5}) = 5 \] Both sets of values satisfy the original system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving systems of equations
Solving systems of equations involves finding values for the variables that satisfy all given equations simultaneously. A 'system' implies multiple equations working together. In this problem, we have a nonlinear system because it includes polynomial terms. The goal is to find pairs of values \( (x, y) \) that work in both equations. Nonlinear systems often need specific methods, like substitution, to solve them effectively. This approach ensures that we systematically reduce the complexity of the equations to find the solution.
Substitution method
The substitution method is an excellent strategy for solving systems of equations, especially when equations are easily rearranged. First, solve one of the equations for one variable. In our example, we solve the second equation \( x - 15y = 5 \) for \( x \) to get \( x = 5 + 15y \). Next, substitute this expression into the other equation. This substitution reduces the system to a single equation in one variable, which can then be solved using other algebraic techniques.
Quadratic formula
The quadratic formula is used to solve equations of the form \( ax^2 + bx + c = 0 \). In the context of our substituted equation, we ended up with \( 75y^2 + 25y - 2 = 0 \). Here, the coefficients are \( a = 75 \), \( b = 25 \), and \( c = -2 \). Using the quadratic formula, \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we substitute our values and solve for \( y \). This formula provides a systematic way to find solutions for quadratic equations, including complex or imaginary solutions if the discriminant (the part under the square root) is negative.
Complex solutions in algebra
Complex solutions arise when solving equations whose solutions are not strictly real numbers. A complex number includes a real part and an imaginary part, written in the form \( a + bi \), where \( i \) is the imaginary unit (\( i^2 = -1 \)). In our problem, all solutions were real numbers. However, understanding complex solutions is crucial because many nonlinear systems or quadratic equations result in complex numbers. Algebraically, solving for complex solutions often involves using the quadratic formula and interpreting the square roots of negative numbers correctly.

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Most popular questions from this chapter

Find the equation of the circle passing through the given points. $$(-5,0),(2,-1), \text { and }(4,3)$$

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Solve each problem. Several years ago, mathematical ecologists created a model to analyze population dynamics of the endangered northern spotted owl in the Pacific Northwest. The ecologists divided the female owl population into three categories: juvenile (up to \(1 \text { yr old }),\) subadult \((1\) to 2 yr old ) and adult (over 2 yr old). They concluded that the change in the makeup of the northern spotted owl population in successive years could be described by the following matrix equation. $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=\left[\begin{array}{rrr} 0 & 0 & 0.33 \\ 0.18 & 0 & 0 \\ 0 & 0.71 & 0.94 \end{array}\right]\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ The numbers in the column matrices give the numbers of females in the three age groups after \(n\) years and \(n+1\) years. Multiplying the matrices yields the following. \(j_{n+1}=0.33 a_{n}\) Each year 33 juvenile females are born for each 100 adult females. \(s_{n+1}=0.18 j_{n}\) Each year 18\% of the juvenile females survive to become subadults. \(a_{n+1}=0.71 s_{n}+0.94 a_{n} \quad\) Each year \(71 \%\) of the subadults survive to become adults, and \(94 \%\) of the adults survive. (a) Suppose there are currently 3000 female northern spotted owls made up of 690 juveniles, 210 subadults, and 2100 adults. Use the matrix equation on the preceding page to determine the total number of female owls for each of the next 5 yr. (b) Using advanced techniques from linear algebra, we can show that in the long run, $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right] \approx 0.98359\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ What can we conclude about the long-term fate of the northern spotted owl? (c) In the model, the main impediment to the survival of the northern spotted owl is the number 0.18 in the second row of the 3 \(\times 3\) matrix. This number is low for two reasons. The first year of life is precarious for most animals living in the wild. In addition, juvenile owls must eventually leave the nest and establish their own territory. If much of the forest near their original home has been cleared, then they are vulnerable to predators while searching for a new home. Suppose that, thanks to better forest management, the number 0.18 can be increased to \(0.3 .\) Rework part (a) under this new assumption.

Solve each problem. A sparkling-water distributor wants to make up 300 gal of sparkling water to sell for \(\$ 6.00\) per gallon. She wishes to mix three grades of water selling for \(\$ 9.00, \$ 3.00,\) and \(\$ 4.50\) per gallon, respectively. She must use twice as much of the S4.50 water as of the \(\$ 3.00\) water. How many gallons of each should she use?
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