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Chapter 1: Problem 74

Solve for the indicated variable. $$ 4+\sqrt{x^{2}+y^{2}}=z \text { for } y $$

Short Answer

Expert verified
y = \pm\sqrt{(z - 4)^2 - x^2}

Step by step solution

01

Isolate the Square Root Term

Subtract 4 from both sides of the equation to isolate the square root term: \[4 + \sqrt{x^2 + y^2} = z\]\[\sqrt{x^2 + y^2} = z - 4\]
02

Square Both Sides

To eliminate the square root, square both sides of the equation:\[(\sqrt{x^2 + y^2})^2 = (z - 4)^2\]\[x^2 + y^2 = (z - 4)^2\]
03

Isolate y^2

Subtract \(x^2\) from both sides of the equation:\[y^2 = (z - 4)^2 - x^2\]
04

Solve for y

Take the square root of both sides, remembering to include the \(\pm\) symbol:\[y = \pm\sqrt{(z - 4)^2 - x^2}\]

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Key Concepts

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

Isolate Variables
In algebra, isolating the variable means manipulating an equation to get the unknown variable by itself on one side of the equation. This is often the first step in solving for a specific variable.
For example, in the given equation: \(4 + \sqrt{x^{2}+y^{2}}=z\), we want to solve for \(y\).
To do this, we start by removing the constants and terms from the side of the equation containing the variable.
In our problem, we subtract 4 from both sides to isolate the square root term:
\[\sqrt{x^2 + y^2} = z - 4\]. Now, the variable term \(\sqrt{x^2 + y^2}\) is isolated.
Square Both Sides
Squaring both sides of an equation is a method used to eliminate a square root. When we have an equation with a square root, squaring helps to simplify it.
In our equation: \(\sqrt{x^2 + y^2} = z - 4\), the next step after isolating the square root term is to square both sides to get rid of the square root:
\[(\sqrt{x^2 + y^2})^2 = (z - 4)^2\].
This simplifies to: \[x^2 + y^2 = (z - 4)^2\].
Notice how squaring both sides removes the square root, making the equation easier to solve.
Algebraic Manipulation
Algebraic manipulation involves using arithmetic operations and algebraic rules to simplify or rearrange equations.
After squaring both sides of our equation: \(x^2 + y^2 = (z - 4)^2\), we need to isolate \(y^2\) to further solve for \(y\).
We subtract \(x^2\) from both sides: \[y^2 = (z - 4)^2 - x^2\].
This is a crucial step where we carefully apply arithmetic operations to manipulate the equation in a way that our target variable (\(y\)) becomes more straightforward to solve.
Square Root Operation
Taking the square root is the final step when an equation involves a squared term. It helps to find the original variable's value before it was squared.
In our simplified equation: \(y^2 = (z - 4)^2 - x^2\), we will now take the square root of both sides to solve for \(y\):
\[y = \pm \sqrt{(z - 4)^2 - x^2}\].
Remember to include the \(\pm\) symbol since both positive and negative values of \(y\) satisfy the equation.
The solution \(y = \pm \sqrt{(z - 4)^2 - x^2}\) gives us the possible values for \(y\) in terms of \(x\) and \(z\).
By understanding each step, it becomes easier to systematically solve algebraic equations.

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