Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation represents \(y\) as a function of \(x.\) $$x=-y+5$$

Step-by-Step Solution

Answer

Yes, the given equation represents \(y\) as a function of \(x\).

Step 1: Rearrange the Given Equation

Take the equation \(x=-y+5\). To make \(y\) the subject, move \(y\) to the left side while keeping the variable \(x\) on the right side. This results in the equation \(y=-x+5\).

Step 2: Analyze the Equation

Look at the rearranged equation \(y=-x+5\). For every value of \(x\), there is exactly one corresponding value of \(y\). The graph of this equation would be a straight line, and for any given \(x\) value, there's a unique \(y\) value, so the vertical line test is passed.

Step 3: Conclusion

Since an input \(x\) will yield exactly one output for \(y\), the equation in the form \(y=-x+5\) represents \(y\) as a function of \(x\).

Key Concepts

Function of x

In algebra, the concept of a function of x, often written as f(x), is foundational. It describes a special relationship where every input value x has a single corresponding output value. This is like saying for each question there is only one answer.

To visualize this, imagine you have a machine that for every number you feed into it, you get out a snack. If you put in a '5', you might get a chocolate bar, and every time you input '5', you always get the same chocolate bar - the output is predictable and consistent. That consistency is what makes this relationship a function.

In the given exercise, when we see equations like the initial one, \(x=-y+5\), it might not be immediately clear if it represents y as a function of x. But after rearranging, we see that indeed for every x we choose, we can find one and only one y. This singularity ensures it's a function of x.

Vertical Line Test

The vertical line test is a visual way to determine if a graph represents a function. Here's how it works: imagine drawing vertical lines (straight up and down) across the graph. If each vertical line touches the graph in exactly one place, then the graph represents a function.

Picture a wall of locker doors. If you can open each locker door without bumping into another (each vertical line), then each locker (x-value) has its own space (y-value), and you've got yourself a function! In contrast, if a vertical line touches the graph in more than one spot, it's like having a key that opens multiple lockers, which means we do not have a function.

In our exercise, once the equation is rearranged to \(y=-x+5\), graphing it would yield a straight line, a kind of graph that will pass the vertical line test with flying colors, confirming that y is a function of x.

Graphing Linear Equations

Graphing is like drawing a map for numbers. When we graph linear equations - equations that make straight lines when graphed - we are translating math into a picture. Every point on the line is a stop on a treasure hunt, where the x marks your step and the y is the treasure waiting for you.

In practice, we usually start with two points to draw the line. Think of it as having two friends standing in different locations. Once you know where they are - by their coordinates (x, y) - you can draw a straight path connecting them, which is your line. Following the rearrangement of our exercise to \(y=-x+5\), if we pick any two values for x, we can solve for their corresponding y-values and plot these two points. Draw a line through them, and voila! You've graphed your linear equation.

Solving for y

Solving for y is like solving a mystery where 'y' is what we need to find out. We usually have clues, which are the other numbers and x in the equation. The process usually involves moving things around in your equation so that y is on one side of the equation and everything else is on the other side.

When we took the exercise's original equation \(x=-y+5\) and rearranged it to get \(y=-x+5\), we were essentially isolating y - it's like finding out 'whodunnit' in a detective story. By getting \(y\) by itself, we can clearly see how the value of \(y\) changes as we try different values for \(x\). This not only helps us in graphing, but also in understanding the very nature of the function we're dealing with.