Which of the following equations represents a linear function?

Which of the following equations represents a linear function?

• A. x^2 + y^2 = 1
• B. y = 2x + 3
• C. xy = 5
• D. y = x^3

Answer: (B)

A linear function is defined by an equation that can be represented in the form \( y = mx + b \), where \( m \) and \( b \) are constants, and \( m \) represents the slope of the line, while \( b \) indicates the y-intercept. The equation \( y = 2x + 3 \) perfectly fits this form. Here, the slope \( m = 2 \) indicates that for every one unit increase in \( x \), \( y \) increases by two units, and the y-intercept \( b = 3 \) tells us that the line crosses the y-axis at \( (0, 3) \). This demonstrates a constant rate of change, which is a key characteristic of linear functions. In contrast, the other equations represent different types of relationships: - The equation \( x^2 + y^2 = 1 \) describes a circle, not a linear function, as it involves squared terms. - The equation \( xy = 5 \) depicts a hyperbola, which also shows a non-linear relationship because \( y \) is implicitly defined in terms of \( x \) and involves multiplicative interaction. - Lastly, the equation

A linear function is defined by an equation that can be represented in the form ( y = mx + b ), where ( m ) and ( b ) are constants, and ( m ) represents the slope of the line, while ( b ) indicates the y-intercept.

The equation ( y = 2x + 3 ) perfectly fits this form. Here, the slope ( m = 2 ) indicates that for every one unit increase in ( x ), ( y ) increases by two units, and the y-intercept ( b = 3 ) tells us that the line crosses the y-axis at ( (0, 3) ). This demonstrates a constant rate of change, which is a key characteristic of linear functions.

In contrast, the other equations represent different types of relationships:

• The equation ( x^2 + y^2 = 1 ) describes a circle, not a linear function, as it involves squared terms.

• The equation ( xy = 5 ) depicts a hyperbola, which also shows a non-linear relationship because ( y ) is implicitly defined in terms of ( x ) and involves multiplicative interaction.

• Lastly, the equation