91Ó°ÊÓ

The absolute value of a number refers to its distance from zero on the number line, regardless of direction. This means that the absolute value is always a non-negative number. For example, the absolute value of both \(-3\) and \(+3\) is \(|3| = 3\).

In mathematics, when you see \(|x|\), it asks "how far is x from 0?" without caring whether it is to the left or right on the number line.

The absolute value function is particularly important when solving equations because it affects how we interpret the \(x\) values. It ensures that no matter the solution direction, the distance is viewed equally. So, even if \(x\) is positive or negative, \(|x|\) neutralizes the sign to focus purely on numerical magnitude.

Linear equations are equations that graph as straight lines. These equations typically have variables raised to the power of 1, like \(y\) and \(x\). A basic structure of a linear equation in two variables is \(y = mx + b\), where \(m\) represents the slope, and \(b\) represents the y-intercept.

In our exercise, we altered the equation \(|x|-y=5\) into a more familiar linear form \(y = |x| - 5\). The graph of this equation would have a V-shape because the \(|x|\) introduces a piecewise linear behavior, causing a sharp 'corner' at \(x = 0\).

In this case, the linear equation shows how \(y\), a dependent variable, changes in direct response to changes in \(x\), an independent variable. The representation reveals that for every \(x\), there is a specific \(y\) value calculated accordingly.

Isolating a variable involves rearranging an equation so that the variable appears alone on one side of the equation. This helps to identify how one variable affects another within the equation.

In our problem, we start with the equation \(|x|-y=5\). We performed isolation by moving terms around to separate \(y\). By subtracting \(|x|\) from both sides, we simplified the equation to \(y = |x| - 5\).

This practice is crucial because it uncovers the role and relationship of variables in the equation, making it easier to solve based on input values. Isolating variables also helps in graphing the equation since it becomes more straightforward to see how \(y\) changes with \(x\). This step is fundamental for defining a function where each input \(x\) maps to a single output \(y\).