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Chapter 2: Problem 33

Determine whether each equation defines \(y\) as a function of \(x\). $$x=3 y-9$$

Short Answer

Expert verified
Yes, y is a function of x because we can express y as \(y = \frac{x + 9}{3}\).

Step by step solution

01

Isolate the variable y

To determine if the equation defines y as a function of x, we need to solve for y in terms of x. Start by isolating y in the given equation: \[x = 3y - 9\]
02

Add 9 to both sides

To isolate y, first add 9 to both sides of the equation: \[x + 9 = 3y\]
03

Divide both sides by 3

Next, divide both sides by 3 to solve for y: \[y = \frac{x + 9}{3}\]
04

Interpret the result

Since we have expressed y in terms of x as \(y = \frac{x + 9}{3}\), this shows that for each value of x, there is exactly one corresponding value of y. Therefore, y is a function of x.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
Isolating a variable is a crucial step when solving equations. It means we want to get the variable by itself on one side of the equation. This process helps us understand how the variables relate to each other.
To isolate a variable:
  • We perform arithmetic operations like addition, subtraction, multiplication, or division.
  • We do the same operation on both sides of the equation to maintain equality.
In the given exercise, we started with the equation \[x = 3y - 9\]\ and isolated y. We first added 9 to both sides \[x + 9 = 3y\],\ and then divided both sides by 3 to get y by itself: \[y = \frac{x + 9}{3}\]. By isolating y, we transformed the original equation and made it easier to analyze.
Solving Linear Equations
Solving linear equations involves finding the value of the variable that makes the equation true. A linear equation is any equation that can be written in the form \[ax + b = c\],\ where a, b, and c are constants. Here are steps to solve linear equations:
  • Isolate the variable by moving constants to the other side of the equation using arithmetic operations.
  • Simplify the equation until you have the variable on one side and a number on the other.
In our exercise, the equation \[x = 3y - 9\] is a linear equation. We followed these steps by first isolating y and then solving for it in terms of x.
Function Definition
A function is a special relationship between two variables such that each input (or x value) corresponds to exactly one output (or y value). We say y is a function of x if for every x, there is only one y. To determine if an equation defines y as a function of x:
  • Isolate y in the equation, if possible.
  • Ensure that each x value gives exactly one y value.
In our sample equation, after isolating y to get \[y = \frac{x + 9}{3}\],\ we see that for each \(x\)\, there is a unique corresponding \(y\).\ This confirms that the equation defines \(y\)\ as a function of \(x\).\ Understanding functions is fundamental in mathematics, as it allows us to see how changing one variable affects another.

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Most popular questions from this chapter

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