Problem 15
Question
State whether each equation or function is linear. Write yes or no. If no, explain your reasoning. \(\frac{1}{x}+3 y=-5\)
Step-by-Step Solution
Verified Answer
No, because \(\frac{1}{x}\) is a nonlinear term.
1Step 1: Identify the standard form of a linear equation
A linear equation in two variables, such as in the form of one of the variables in terms of the others (e.g., \(y = mx + c\)), can also be represented as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants and \(x\) and \(y\) are variables.
2Step 2: Analyze given equation for linearity
The given equation is \(\frac{1}{x} + 3y = -5\). For it to be linear, each term must either be a constant or a constant multiplied by \(x\) or \(y\) – this means only terms of the first degree in \(x\) and \(y\) are allowed.
3Step 3: Assess the term \(\frac{1}{x}\)
The term \(\frac{1}{x}\) is not a constant nor a simple multiplication of a variable with a constant, as it involves a division by the variable \(x\). This results in a nonlinear transformation, disqualifying the equation as linear.
4Step 4: Conclusion based on assessment
Since the term \(\frac{1}{x}\) appears in the equation, it causes the equation to violate the necessary structure of a linear equation. Therefore, the equation is not linear.
Key Concepts
Understanding Nonlinear EquationsWhat is the Standard Form of an Equation?Understanding Variables in Equations
Understanding Nonlinear Equations
A nonlinear equation is one in which the variables are not simply raised to the power of 1, i.e., they do not cap at linear exponents. Instead, nonlinear equations can involve squared terms, radicals, fractions with variables in the denominator, and more.
For example, in the equation \( \frac{1}{x} + 3y = -5 \), the presence of \( \frac{1}{x} \) indicates a nonlinear component because the variable \( x \) is in the denominator.
Such expressions mean the relationship between the variables \( x \) and \( y \) is not a straight line when graphed. Nonlinear equations can form curves or other complex shapes.
For example, in the equation \( \frac{1}{x} + 3y = -5 \), the presence of \( \frac{1}{x} \) indicates a nonlinear component because the variable \( x \) is in the denominator.
Such expressions mean the relationship between the variables \( x \) and \( y \) is not a straight line when graphed. Nonlinear equations can form curves or other complex shapes.
- Variables with powers other than 1
- Radical expressions of variables
- Products of variables
What is the Standard Form of an Equation?
The standard form is a conventional way of writing equations down so that they speak a common language that can be universally understood.
For linear equations, this standard form is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. This clean structure makes it simpler to analyze and solve equations.
In contrast, our example equation \( \frac{1}{x} + 3y = -5 \) cannot be arranged into this standard linear form due to the term \( \frac{1}{x} \).
Equations that deviate from the standard form, particularly by having higher degree terms or variables in the denominator, denote non-linear patterns. Remember:
For linear equations, this standard form is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. This clean structure makes it simpler to analyze and solve equations.
In contrast, our example equation \( \frac{1}{x} + 3y = -5 \) cannot be arranged into this standard linear form due to the term \( \frac{1}{x} \).
Equations that deviate from the standard form, particularly by having higher degree terms or variables in the denominator, denote non-linear patterns. Remember:
- Standard form helps to quickly verify linearity.
- It simplifies comparison and computation of equations.
- Not all equations can be reshaped into the standard form of linear equations.
Understanding Variables in Equations
Variables are the symbols used to represent unknown values in mathematical equations which can change depending on the conditions of the problem.
In the given equation \( \frac{1}{x} + 3y = -5 \), \( x \) and \( y \) are not only placeholders for specific numbers, they represent elements that influence the outcome of the equation.
The interplay of variables determines whether an equation is linear or nonlinear. In a linear setting, variables are typically found in the first power.
In contrast, our equation has \( x \) in the denominator, suggesting dependency that is more complex than a simple, singular, first power relationship. Thus, the relationship between \( x \) and \( y \) is affected heavily by changes in \( x \), making the equation nonlinear. Here are some key points about variables in equations:
In the given equation \( \frac{1}{x} + 3y = -5 \), \( x \) and \( y \) are not only placeholders for specific numbers, they represent elements that influence the outcome of the equation.
The interplay of variables determines whether an equation is linear or nonlinear. In a linear setting, variables are typically found in the first power.
In contrast, our equation has \( x \) in the denominator, suggesting dependency that is more complex than a simple, singular, first power relationship. Thus, the relationship between \( x \) and \( y \) is affected heavily by changes in \( x \), making the equation nonlinear. Here are some key points about variables in equations:
- Symbols for unknowns or varying values.
- Central in assessing equation types.
- Dictate the complexity of equations based on their placement and degree.
Other exercises in this chapter
Problem 15
Write an equation in slope-intercept form for the line that satisfies each set of conditions. slope \(-\frac{1}{2},\) passes through \((1,3)\)
View solution Problem 15
Find the slope of the line that passes through each pair of points. $$ (8,7),(7,-6) $$
View solution Problem 16
Graph each inequality. $$ y \geq 1 $$
View solution Problem 16
Write an equation in slope-intercept form for the line that satisfies each set of conditions. slope \(\frac{3}{2}\) passes through \((-5,1)\)
View solution