Problem 113 - Introductory Chemistry Atoms First

Question

(a) Solve the equation \(y=z / x\) for \(x\). (b) Solve the equation \(y=z / 2 x\) for \(x\).

Step-by-Step Solution

Answer

(a) \( x = \frac{z}{y} \) (b) \( x = \frac{z}{2y} \)

Step 1: (a) Step 1: Rearrange the equation to isolate x in terms of y and z

We are given the equation: \( y = \frac{z}{x} \) To isolate x, we can multiply both sides of the equation by x, which gives: \( xy = z \)

Step 2: (a) Step 2: Solve for x in terms of y and z

Now we need to continue to manipulate this equation to solve for x. Since y is being multiplied by x, we can divide both sides of the equation by y, resulting in: \( x = \frac{z}{y} \)

Step 3: (b) Step 1: Rearrange the equation to isolate x in terms of y and z

We are given the equation: \( y= \frac{z}{2x} \) To isolate x, we can multiply both sides of the equation by 2x, which gives: \( 2xy = z \)

Step 4: (b) Step 2: Solve for x in terms of y and z

Now we need to continue to manipulate this equation to solve for x. Since 2y is being multiplied by x, we can divide both sides of the equation by 2y, resulting in: \( x = \frac{z}{2y} \)

Key Concepts

Equation Solving

Equation solving is all about finding an unknown value, typically shown by a variable like \( x \), in an equation. When solving equations, the goal is to perform operations that simplify the equation to isolate this variable. For example, given \( y = \frac{z}{x} \), the aim is to rearrange and manipulate the equation so that we express \( x \) in terms of \( y \) and \( z \). Remember, whatever you do to one side of the equation, you must do to the other to keep the balance. This can involve operations like multiplying, dividing, adding, or subtracting terms.

Rearranging Equations

Rearranging equations is crucial for solving problems and involves altering the equation structure to get a desired outcome. Let's look at rearranging \( y = \frac{z}{x} \). The initial step is to eliminate the fraction by multiplying both sides by \( x \), resulting in \( xy = z \). This process helps to simplify complex expressions. Rearranging is useful beyond mathematics, such as in physics to find force or energy. You can break down any rearrangement by following these steps:

• Identify the term you are solving for.
• Perform operations to simplify the structure.
• Ensure every operation maintains balance by doing it to both sides.

The key is to make the equation look simpler, clearer, and more focused on what you need to find.

Isolating Variables

Isolating a variable refers to manipulating an equation such that the variable stands alone on one side of the equation. For instance, in \( y = \frac{z}{x} \), isolating \( x \) means transforming the equation through multiplication to get \( xy = z \), then dividing by \( y \) to get \( x = \frac{z}{y} \). The process involves:

• Identifying which variable needs to be isolated.
• Using inverse operations to "undo" other operations, like using division to cancel multiplication.
• Achieving the simplest form where the chosen variable is alone on one side.

The goal is to reorganize the equation such that it provides a direct answer for the variable, making it easier to substitute numerical values later if needed.