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Chapter 2: Problem 113

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

Short Answer

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

Step by step solution

01

(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: \( xy = z \)
02

(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: \( x = \frac{z}{y} \)
03

(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: \( 2xy = z \)
04

(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: \( x = \frac{z}{2y} \)

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Key Concepts

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

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.

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Most popular questions from this chapter

Indicate whether the trailing zero in each value is significant, not significant, or possibly significant: (a) \(540 \pm 0.5\) (b) \(540 \pm 5\) (c) \(0.540\) (d) \(0.000540\) (e) 540

Do the following calculations and express each answer in scientific notation: (a) \(\left(5.03 \times 10^{2}\right)+\left(8.1 \times 10^{1}\right)\) (b) \(\left(8.32 \times 10^{-5}\right) \times\left(0.53 \times 10^{4}\right)\) (c) \(\frac{6.02 \times 10^{23}}{3}\), where the 3 is an exact number (d) \(\left(3.960 \times 10^{3}\right)-\left(4.62 \times 10^{2}\right)\)

A train traveling at \(45.0\) miles \(/ \mathrm{h}\) has to make a trip of \(100.0\) miles. How many minutes will the trip take? Use unit analysis to calculate your answer, and show your work.

If the same amount of heat energy is added to a beaker containing \(100 \mathrm{~mL}\) of ethanol and a beaker containing \(100 \mathrm{~mL}\) of water, which liquid experiences the greater rise in temperature?

For solids, the amount of material per unit volume is often expressed in grams per milliliter, whereas for gases the amount of material per unit volume is usually expressed in grams per liter. If the amount of matter in air is \(1.34 \mathrm{~g} / \mathrm{L}\), what is this value in: (a) \(\mathrm{g} / \mathrm{mL}\) (b) \(\mathrm{kg} / \mathrm{L}\) (c) \(\mathrm{kg} / \mathrm{mL}\) ?
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