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Chapter 2: Problem 21

For each formula, express y as a function of x. $$\ x-y=4$$

Short Answer

Expert verified
y = x - 4

Step by step solution

01

Start with the given equation

The original equation is x - y = 4 . We need to manipulate this equation to solve for y.
02

Isolate the term -y

Subtract x from both sides to isolate the term -y: -y = 4 - x .
03

Multiply by -1

To solve for y, multiply both sides of the equation by -1 to get y = -4 + x or y = x - 4 .

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Key Concepts

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

isolate variable
When solving linear equations, one key concept is to isolate the variable you are solving for. In this example, the equation is given as \(x - y = 4\). Our goal is to express \(y\) as a function of \(x\).
We start by isolating \(-y\) . This means we need to move \(x\) to the other side of the equation.
To do this, we subtract \(x\) from both sides.
Now, the equation looks like this: \(-y = 4 - x\) . By isolating the term containing \(y\), we have made the problem simpler.
manipulate equations
Another important skill is learning how to manipulate equations. This means using mathematical operations to transform the equation into an easier form.
In our example, we have \(-y = 4 - x\) . To make \(y\) positive, we multiply both sides of the equation by -1.
When we do this, keep in mind the rule that multiplying or dividing both sides of an equation by the same number does not change the equality:
  • Multiplying \(-y\) by -1 gives us \(y\).
  • Multiplying \(4 - x\) by -1 changes the signs of each term, resulting in \(-4 + x\).
The equation now becomes: \(y = -4 + x\). This step is crucial in solving for the variable.
solving for y
The final step in expressing \(y\) as a function of \(x\) is to simply write the answer clearly.
In our manipulated equation, we have \(y = -4 + x\).
This can also be written as \(y = x - 4\), which is just a rearrangement of terms. Both forms are correct and acceptable, so it does not matter which one you use.
Always make sure your final answer clearly shows \(y\) on the left side and everything else on the right side.
This makes it easy to see that \(y\) is indeed expressed as a function of \(x\).
Solving for a variable often involves isolation and manipulation, and writing the final clear solution.

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