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Chapter 9: Problem 19

Solve the given nonlinear system. $$ \left\\{\begin{array}{l} x y=1 \\ x+y=1 \end{array}\right. $$

Short Answer

Expert verified
The system has no real solutions.

Step by step solution

01

Express y from the Second Equation

Start by expressing one variable (e.g., \(y\)) in terms of the other variable (\(x\)) using the second equation. From \(x + y = 1\), we have \(y = 1 - x\).
02

Substitute Expression into First Equation

Substitute the expression for \(y\) into the first equation \(xy = 1\). This gives \(x(1 - x) = 1\). Simplifying yields the quadratic equation \(x - x^2 = 1\).
03

Rearrange the Quadratic Equation

Rearrange the equation \(x - x^2 = 1\) to standard quadratic form, which results in \(x^2 - x + 1 = 0\).
04

Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -1\), and \(c = 1\). Calculating the discriminant, \(b^2 - 4ac = (-1)^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3\), which is negative. This indicates no real solutions.

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Key Concepts

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

Quadratic Equation
A quadratic equation is a vital part of algebra and can be identified by its standard form: \( ax^2 + bx + c = 0 \). This type of equation is called 'quadratic' because the highest power of the variable \( x \) is 2. Understanding and solving quadratic equations is crucial because they frequently appear in various mathematical contexts.
  • **Key Elements**: The quadratic equation will typically involve constants \( a \), \( b \), and \( c \), where \( a \) is not equal to zero.
  • **Graph Shape**: The graph of a quadratic equation is a parabola, which can open either upwards or downwards depending on the value of \( a \).
  • **Solving Quadratics**: Solutions to quadratic equations can be found by factoring, completing the square, or using the quadratic formula.
In our specific problem, we derived the quadratic equation \( x^2 - x + 1 = 0 \) by substituting one variable in terms of the other. This equation helps us explore potential solutions within the context of the initial nonlinear system.
Discriminant
The discriminant is a key concept linked to quadratic equations. It reveals fundamental insights about the nature of solutions. For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is given by \( b^2 - 4ac \).
  • **Importance**: The discriminant helps determine the number and type of roots (solutions) without actually solving the equation.
  • **Interpretation**:
    • If \( b^2 - 4ac > 0 \), there are two distinct real roots.
    • If \( b^2 - 4ac = 0 \), there is one real and repeated root.
    • If \( b^2 - 4ac < 0 \), there are two complex roots, implying no real solutions.
In our example, the calculated discriminant was \(-3\), indicating that the quadratic equation has no real solutions. This means the original nonlinear system does not have a real solution set for \( x \) and \( y \).
Substitution Method
The substitution method is a common technique used to solve systems of equations, including nonlinear ones. It involves replacing a variable in one equation with an expression obtained from another equation.
  • **Basic Steps**:
    • First, solve one of the equations for one of the variables.
    • Then substitute this expression into the other equation.
    • Solve the resulting equation.
    • Use the solution to find the value of the other variable.
  • **Benefits**: This method simplifies the system, making it easier to identify solutions by working with one variable at a time.
  • **Limitations**: It might become complicated if the expressions get too complex or when dealing with many variables.
In our exercise, we used the substitution method to express \( y \) in terms of \( x \) and substitute it back into the other equation. This effectively reduced the nonlinear system into a single quadratic equation for further analysis.

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Most popular questions from this chapter

Number of Gallons A 100-gal tank is full of water in which \(50 \mathrm{lb}\) of salt is dissolved. A second tank contains 200 gal of water with \(75 \mathrm{lb}\) of salt. How much should be removed from both tanks and mixed together in order to make a solution of 90 gal with \(\frac{4}{9} \mathrm{lb}\) of salt per gallon?

Solve the given nonlinear system. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=5 \\ y=x^{2}-5 \end{array}\right. $$

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{2 x^{2}-3}{x^{3}+x^{2}} $$

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{4}{x^{3}\left(x^{2}+1\right)} $$

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{aligned} -5 x+y+z &=0 \\ 4 x-y &=0 \\ 2 x-y+2 z &=0 \end{aligned}\right. $$
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