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Chapter 1: Problem 73

Solve for the indicated variable. $$ 16+\sqrt{x^{2}-y^{2}}=z \text { for } x $$

Short Answer

Expert verified
x = 卤鈭�((z-16)^2 + y^2)

Step by step solution

01

Isolate the Square Root Term

Subtract 16 from both sides of the equation: \[ \text{16} + \text{ }\boldsymbol{ \text{鈭殅\text{x^2-y^2}}= \text{z} \] \[ \text{鈭殅\text{x^2-y^2}= \text{z-16} \]
02

Square Both Sides

Eliminate the square root by squaring both sides of the equation: \[ (\text{鈭殅\text{x^2-y^2})^2= (\text{z-16})^2 \] Which simplifies to: \[ \text{x^2-y^2}= (\text{z-16})^2 \]
03

Isolate x^2

Add \( y^2 \) to both sides to isolate \( x^2 \): \[ \text{x^2} = (\text{z-16})^2 + \text{y^2}\]
04

Solve for x

Take the square root of both sides to solve for \( x \): \[ \text{x} = \text{卤鈭殅\big((\text{z-16})^2 + y^2\big)\]

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Key Concepts

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

Isolate the Variable
In algebra, isolating the variable you need to solve for is often the first key step. Here, we need to solve for \(x\) in the equation:
16 + \( \text{鈭殅\text{x^2 - y^2} = z \).
To isolate the square root term, subtract 16 from both sides:
\( \text{鈭殅\text{x^2 - y^2} = z - 16 \).
This step helps us focus on isolating \(x\) by getting rid of any constants or other variables.
Square Root
The square root term in the equation presents a critical challenge. To eliminate the square root, we square both sides of the equation:
\( (\text{鈭殅\text{x^2 - y^2})^2 = (z-16)^2 \).
This simplifies to:
\( x^2 - y^2 = (z-16)^2 \).
This step clears the square root, making the equation easier to solve.
Algebraic Manipulation
Now, we need further algebraic manipulation to solve for \(x\). Add \(y^2\) to both sides:
\( x^2 = (z-16)^2 + y^2 \).
This operation isolates \(x^2\), making it clear what step we need to perform next. Always maintain equality by performing identical operations on both sides.
Quadratic Equation
Finally, we solve the quadratic equation by taking the square root of both sides:
\( x = \text{卤鈭殅((z-16)^2 + y^2) \).
This final step gives both positive and negative roots, because squaring either a positive or negative number yields the same result. Always remember to include \( \text{卤} \) when solving quadratic equations involving square roots.

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