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Chapter 2: Problem 4

Determine which equations are linear equations in the variables \(x, y,\) and \(z .\) If any equation is not linear, explain why not. $$2 x-x y-5 z=0$$

Short Answer

Expert verified
The equation is not linear because of the term \(-xy\).

Step by step solution

01

Analyze the Terms

Let's break down each term in the equation \(2x - xy - 5z = 0\) to determine if they are linear. In a linear equation, each term must be either a constant or a product of a constant and a single variable raised to the first power.
02

Check Each Term for Linear Qualities

- The term \(2x\) is linear in \(x\) since it is a constant (2) multiplied by \(x\) raised to the power of 1.- The term \(-5z\) is linear in \(z\) because it is a constant (-5) multiplied by \(z\) raised to the power of 1.- The term \(-xy\) is *not* linear. It involves the product of two variables \(x\) and \(y\), which makes it nonlinear.
03

Conclusion About Linearity

Since the equation \(2x - xy - 5z = 0\) includes the term \(-xy\), it is not linear. A linear equation cannot have any products or higher powers of its variables.

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Key Concepts

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

Linearity Analysis
Linearity analysis helps us understand whether an equation is linear or not. A linear equation is one where each term satisfies specific rules:
  • Each term is either a constant or a constant multiplied by a variable.
  • Variables should not be raised to any power other than one.
  • There should be no products of different variables within a single term.
In the exercise given, analyzing the linearity of the equation means checking each term separately. If all terms comply with the above rules, the equation is linear. This process of linearly analyzing terms is crucial to identify the true nature of the equation.
Constant and Variable Product
In linear equations, understanding the product of constants and variables is essential. A term is considered linear if it is a simple multiplication of a constant (a fixed number) and a variable (like \(x, y,\) or \(z\)) raised to the first power.
To illustrate:
  • The term \(2x\) involves multiplying the constant 2 by the variable \(x\). Since \(x\) is raised to the first power, \(2x\) is a linear term.
  • Similarly, \(-5z\) is linear because it consists of the constant -5 multiplied by the variable \(z\) to the first power.
Thus, any term that matches the pattern of a constant and variable in this way remains within the bounds of linearity.
Nonlinear Terms
Nonlinear terms do not comply with the rules set for linearity. They typically involve:
  • Variables raised to powers other than one.
  • The product of two or more different variables.
In the equation provided, the term \(-xy\) is identified as nonlinear. Here鈥檚 why:
  • It is a product of two variables, \(x\) and \(y\), which violates the rule of having only a single variable per term.
Understanding nonlinear terms is crucial because even a single nonlinear term can make the entire equation nonlinear. Identifying these terms in an equation helps in evaluating whether the equation qualifies as linear or nonlinear.

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Most popular questions from this chapter

Suppose the coal and steel industries form an open economy. Every \(\$ 1\) produced by the coal industry requires \(\$ 0.15\) of coal and \(\$ 0.20\) of steel. Every \(\$ 1\) produced by steel requires \(\$ 0.25\) of coal and \(\$ 0.10\) of steel. Suppose that there is an annual outside demand for \(\$ 45\) million of coal and \(\$ 124\) million of steel. (a) How much should each industry produce to satisfy the demands? (b) If the demand for coal decreases by \(\$ 5\) million per year while the demand for steel increases by \$6 million per year, how should the coal and steel industries adjust their production?

In general, what is the elementary row operation that "undoes" each of the three elementary row operations \(R_{i} \leftrightarrow R_{j}, k R_{i}\) and \(R_{i}+k R_{j}^{?}\)

Set up and solve an appropriate system of linear equations to answer the questions. The process of adding rational functions (ratios of polynomials by placing them over a common denominator is the analogue of adding rational numbers. The reverse process of taking a rational function apart by writing it as a sum of simpler rational functions is useful in several areas of mathematics; for example, it arises in calculus when we need to integrate a rational function and in discrete mat hematics when we use generating functions to solve recurrence relations. The decomposition of a rational function as a sum of partial fractions leads to a system of linear equations. Find the partial fraction decomposition of the given form. (The capitall) letters denote constants. $$\frac{x^{2}-3 x+3}{x^{3}+2 x^{2}+x}=\frac{A}{x}+\frac{B}{x+1}+\frac{C}{(x+1)^{2}}$$

Compute the first four iterates, using the zero vector as the initial approximation, to show that the Gauss-Seidel method diverges. Then show that the equations can be rearranged to give a strictly diagonally dominant coefficient matrix, and apply the Gauss-Seidel method to obtain an approximate solution that is accurate to within 0.001. $$\begin{aligned}x_{1}-4 x_{2}+2 x_{3} &=2 \\\2 x_{2}+4 x_{3} &=1 \\\6 x_{1}-x_{2}-2 x_{3} &=1\end{aligned}$$

Apply Jacobis method to the given system. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each variable. In each case, compare your answer with the exact solution found using any direct method you like. $$\begin{aligned}4.5 x_{1}-0.5 x_{2} &=1 \\\x_{1}-3.5 x_{2} &=-1\end{aligned}$$
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