To determine which of the given equations is not a linear equation, we first need to understand the definition of a linear equation. A linear equation is one in which the variables appear to the first power and are not multiplied by each other.
Let's analyze each given equation:
A) \( x = 5 \)
This equation represents a situation where \( x \) is equal to a constant. It is a linear equation because \( x \) is to the first power, and it satisfies the general form \( Ax + B = 0 \) (where \( x \) is a variable, and \( A \) and \( B \) are constants).
B) \( y = x + 6 \)
This equation can be rearranged in the form \( y - x - 6 = 0 \), which fits the general linear equation form \( Ax + By + C = 0 \) (where \( x \) and \( y \) are variables, and \( A \), \( B \), and \( C \) are constants). Both \( x \) and \( y \) appear to the first power and are not multiplied together. Therefore, this is also a linear equation.
C) \( 2x = 3y \)
This equation can be rearranged as \( 2x - 3y = 0 \), which again fits the general linear form \( Ax + By + C = 0 \) (with \( A = 2 \), \( B = -3 \), and \( C = 0 \)). Both \( x \) and \( y \) are still to the first power and are not multiplied by each other. Hence, this is a linear equation.
D) \( y^2 = x^2 + 3 \)
This equation contains \( y \) and \( x \) variables raised to the power of 2. This means that the variables are not to the first power, which violates the condition for a linear equation. Because there are squared terms (\( y^2 \) and \( x^2 \)), this equation is not a linear equation.
To summarize, the given equations are analyzed as follows:
- A) Linear
- B) Linear
- C) Linear
- D) Not Linear
Therefore, the equation \( y^2 = x^2 + 3 \) (option D) is not a linear equation. Thus, the correct answer is:
D) [tex]\( y^2 = x^2 + 3 \)[/tex]
