91Ó°ÊÓ

Isolating variables in algebraic equations is a fundamental skill. It involves re-arranging the equation so that one variable is on one side of the equals sign, and all the other terms are on the other side. This process makes it much easier to understand and solve for the unknown quantity. Let's look at how we isolate variables using our given exercise.

To isolate the variable y, you subtract terms not involving y from both sides of the equation to get y alone on one side. In the provided equation, we start by moving the term involving x to the opposite side, obtaining \(4y = -3\sqrt{x}\). The next step is to eliminate the coefficient of y by dividing both sides by 4, resulting in \(y = \frac{-3\sqrt{x}}{4}\). This process of isolating y gives us a clearer view of the relationship between y and x. However, through this isolation, we discover that the equation is not linear, as it includes a square root function of x, which affects the linearity of the equation.

The slope-intercept form is a way of writing an equation of a line so that the slope (rate of change) and the y-intercept (the value where the line crosses the y-axis) are immediately apparent. This form is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.

When we have an equation like the one given in the exercise, we seek to rewrite it in slope-intercept form to determine if it's a linear function. If we can express our equation with variables x and y such that y is a multiple of x plus some constant, then we have a linear equation. However, if the equation includes powers, roots, or products of x that do not yield a straight line when graphed, the equation cannot be a linear function, as is the case with our exercise example.

Non-linear equations represent relationships between variables that do not form straight lines when graphed. These can include quadratic functions, exponential functions, and as we've seen in our exercise, functions that include square roots. In the given problem, the presence of \(\sqrt{x}\) is a clear indicator of non-linearity.

Linear equations have an important characteristic: their graph is always a straight line, meaning the rate of change between the variables is constant. With non-linear equations, this rate of change is not constant, which results in curves on the graph. The equation \(y = \frac{-3\sqrt{x}}{4}\) from the exercise, therefore, defines a non-linear function of x, as the square root of x results in a curve, which continually changes its slope, rather than maintaining the constant slope that characterizes linear functions.