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When studying linear equations, the concept of variables is fundamental. A variable is a symbol, often a letter like x or y, that stands in for an unknown number. In linear equations, variables appear in their simplest form – they are not raised to any power other than one, and they are not under a radical or in the denominator. In the given example, \( (\cos 3) x + y = -16 \) , x and y are the variables. They represent numbers that can change or vary.

When writing or analyzing linear equations, it's important to ensure the variables meet certain criteria to maintain linearity. This means each variable must only be multiplied by a coefficient (in this case, \( \cos 3 \) for x) and not by each other. They are presented as \( ax + by = c \), where a, b, and c are constants. This form shows that each variable impacts the value of the equation independently.

To determine if an equation is linear, we need to check for certain characteristics. A linear equation will have one or both of the following features: the highest power (degree) of the variables is one, and the variables do not multiply or divide each other or themselves. In the given exercise, the equation \( (\cos 3) x + y = -16 \) is analyzed to check for these features.

The equation maintains its linearity because each variable is of the first degree. This implies that there are no squares, cubes, or higher degree terms of x or y. The constants (like \(\cos 3\)) do not affect the linearity unless they form a product with variables. Since x and y appear in a linear combination without violating these rules, the equation qualifies as linear. This information helps us graph the equation as a straight line and apply various algebraic principles designed specifically for linear models.

The degree of a variable in an equation represents the highest power (exponent) to which the variable is raised. In a linear equation, the degree of each variable must be one. For example, in the equation \( (\cos 3) x + y = -16 \) examined here, both x and y are in the first degree, signified by their lack of exponents, or understood to be raised to the power of one.

If variables have a degree higher than one, such as x^2 or y^3, the equation becomes nonlinear and describes a curve, not a straight line. Variables in linear equations also cannot multiply each other, as in xy, nor can they be in the denominator, like \( \frac{1}{x} \), as this would increase the complexity of the equation. Keeping the variables to the first degree is crucial for the simplicity and predictability of linear equations, which allows them to be solved and graphed more easily.