Determine whether each equation defines y as a function of x. 2x + y = 8

Verified step by step guidance

1. Rewrite the equation in terms of \( y \) to see if it can be expressed as a function of \( x \).
2. Start with the given equation: \( 2x + y = 8 \).
3. Isolate \( y \) by subtracting \( 2x \) from both sides: \( y = 8 - 2x \).
4. Check if for every \( x \) there is exactly one \( y \).
5. Since \( y = 8 - 2x \) is a linear equation, it defines \( y \) as a function of \( x \).

Key Concepts

Function Definition

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if an equation defines y as a function of x, we must check if for every x, there is a unique y. This means that no x-value can produce multiple y-values.

Vertical Line Test

The vertical line test is a visual way to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the relation is not a function. This test helps to quickly assess the uniqueness of y-values for given x-values.

Solving for y

To analyze whether an equation defines y as a function of x, we often rearrange the equation to solve for y explicitly. In the case of 2x + y = 8, isolating y gives y = 8 - 2x, which shows that for each x, there is a corresponding unique y, confirming that it is indeed a function.