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Chapter 7: Problem 26

Solve each formula for the specified variable. $$ S=\frac{H}{m\left(t_{1}-t_{2}\right)} ; H $$

Short Answer

Expert verified
H = S \cdot m(t_1 - t_2)

Step by step solution

01

Understand the given formula

The formula given is \[S = \frac{H}{m(t_1 - t_2)}\]and you need to solve for the variable \(H\).
02

Isolate the variable H

To isolate \(H\), multiply both sides of the equation by \(m(t_1 - t_2)\) to eliminate the denominator. This gives:\[S \cdot m(t_1 - t_2) = H\]
03

Simplify the equation

Now, combine the terms on the left-hand side to make sure that \(H\) is isolated:\[H = S \cdot m(t_1 - t_2)\]

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Key Concepts

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

Isolating Variables
Isolating variables is a crucial skill in algebra. It helps you find the value of a specific variable in an equation. In our example, we need to isolate the variable \(H\). To do this, you need to get \(H\) by itself on one side of the equation. Let's break it down.
In the equation \(S = \frac{H}{m(t_1 - t_2)}\), we see that \(H\) is in the numerator of a fraction on the right-hand side. Here鈥檚 a simple method to isolate \(H\):
  • Multiply both sides by the denominator \(m(t_1 - t_2)\) to cancel it out.
This step changes our equation to \(S \times m(t_1 - t_2) = H\). Now, \(H\) is isolated and by itself on one side!
Isolating a variable often involves these steps:
  • Identify the variable you need to isolate.
  • Use operations like multiplication, division, addition, or subtraction to move other terms to the opposite side.
  • Simplify the equation.
With practice, isolating variables will become second nature.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to solve for a variable. It's like solving a puzzle! For our formula \(S = \frac{H}{m(t_1 - t_2)}\), we need to manipulate it to solve for \(H\).
Step-by-step process:
  • We start with \(S = \frac{H}{m(t_1 - t_2)}\).
  • Multiply both sides by the denominator, \(m(t_1 - t_2)\), to eliminate the fraction: \(S \times m(t_1 - t_2) = H\).
  • This leaves us with \(H = S \times m(t_1 - t_2)\).
By following these steps, we have successfully applied algebraic manipulation. This principle can be used for other equations as well. Remember, keep your work neat to avoid mistakes, and always double-check your steps.
Mathematical Formulas
Mathematical formulas are equations that express relationships between different quantities. They are tools used to solve problems by plugging in known values to find the unknown ones. In our example, the formula given is \(S = \frac{H}{m(t_1 - t_2)}\), which likely represents a physical or scientific relationship. When solving such formulas:
  • Identify what each variable represents.
  • Determine which variable you need to solve for (in this case, \(H\)).
  • Use algebraic manipulation techniques to isolate and solve for that variable.
Understanding formulas also involves knowing units and context. For example, \(S\) might be speed, \(H\) could be distance, and \(m(t_1 - t_2)\) could encompass time factors.
Solving formulas becomes easier with practice. Each step builds your confidence and skill in handling more complex equations and real-life problems.

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