Question

Justify whether the equation y =9x-4
defines a linear or nonlinear function.

Which statement is true?
The function is linear because the equation has the form
.
The function is linear because the equation contains only whole numbers.
The function is nonlinear because the equation does not have the form
.
The function is nonlinear because the equation contains only perfect squares.

Answer 1

To determine whether the equation \( y = 9x - 4 \) defines a linear or nonlinear function, we can analyze its form.

1. A linear function can be expressed in the standard form \( y = mx + b \), where \( m \) and \( b \) are constants. In this case, \( y = 9x - 4 \) fits this format, where \( m = 9 \) and \( b = -4 \).

2. The function is linear because the highest power of the variable \( x \) is 1. This indicates a straight line when graphed.

Now, let's evaluate each of the provided statements:

1. **The function is linear because the equation has the form**: This statement is true. The equation \( y = 9x - 4 \) is in the form of a linear equation.

2. **The function is linear because the equation contains only whole numbers**: This statement is false. The function can be linear regardless of whether it contains whole numbers or not.

3. **The function is nonlinear because the equation does not have the form**: This statement is false. The equation indeed has the correct linear form.

4. **The function is nonlinear because the equation contains only perfect squares**: This statement is false. The equation does not contain perfect squares; it is linear.

Therefore, the correct conclusion is that "The function is linear because the equation has the form \( y = mx + b \)."