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Chapter 6: 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 \times m(t_{1} - t_{2})

Step by step solution

01

- Understand the given formula

The given formula is \( S = \frac{H}{m(t_{1} - t_{2})} \). You are asked to solve for the variable \( H \). This means you need to isolate \( H \) on one side of the equation.
02

- Eliminate the fraction by multiplying both sides

Multiply both sides of the equation by \( m(t_{1} - t_{2}) \) to get rid of the fraction: \( S \times m(t_{1} - t_{2}) = \frac{H}{m(t_{1} - t_{2})} \times m(t_{1} - t_{2}) \). This simplifies to: \( S \times m(t_{1} - t_{2}) = H \).
03

- Rewrite the equation

The equation after simplification is: \( H = S \times m(t_{1} - t_{2}) \). Thus, \( H \) is isolated and the formula is solved for \( H \).

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Key Concepts

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

isolating a variable
Isolating a variable is a fundamental skill in algebra. It involves rearranging an equation so that the variable you are solving for is alone on one side of the equation.
For example, in the given exercise, the goal is to solve the formula for the variable \(H\). This means you need to manipulate the equation so that \(H\) appears by itself on one side of the equation.
Here's a clearer breakdown:
  • Starting with the equation: \( S = \frac{H}{m(t_{1} - t_{2})} \)
  • Recognizing that to isolate \(H\), you need to eliminate the fraction: \(m(t_{1} - t_{2})\) is in the denominator with \(H\) in the numerator.
  • Multiplying both sides of the equation by \(m(t_{1} - t_{2})\) removes the fraction, resulting in: \( S \times m(t_{1} - t_{2}) = H \)
Now, \(H\) is isolated. Practicing this concept with various examples helps solidify your understanding of isolating a variable.
algebraic manipulation
Algebraic manipulation is the technique of rearranging and simplifying algebraic expressions. It requires understanding various algebraic rules and operations.
To solve for a variable, you perform operations that simplify the equation step-by-step without changing its equality.
Let's apply this to the exercise:
  • We start with: \( S = \frac{H}{m(t_{1} - t_{2})} \)
  • Our goal is to clear the fraction, so we multiply both sides by \( m(t_{1} - t_{2}) \), leading to: \( S \times m(t_{1} - t_{2}) = H \)
  • This gives: \( S \times m(t_{1} - t_{2}) = H \)
In algebraic manipulation, always perform the same operation on both sides of the equation to maintain equality.
Techniques such as distributing, combining like terms, and factoring are commonly used during manipulation.
formula rearrangement
Formula rearrangement involves changing the form of a formula to solve for a different variable. This is often needed in subjects like physics, chemistry, and engineering.
Understanding the relationship between variables helps make rearrangement easier.
In the given problem, the original formula is \( S = \frac{H}{m(t_{1} - t_{2})} \).
To rearrange to solve for \( H \):
  • Identify the variable you need to isolate. In this case, \(H\).
  • Determine the operations applied to your variable. Here, \(H\) is divided by \(m(t_{1} - t_{2})\).
  • Use inverse operations to isolate the variable: multiply both sides by \(m(t_{1} - t_{2})\).
  • Rewriting the equation, we get: \( H = S \times m(t_{1} - t_{2}) \)
Formula rearrangement helps transform complex equations into a more useful format. It aids in understanding the relationships and dependencies between different variables within an equation.

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Most popular questions from this chapter

Find an equation of variation in which: \(y\) varies jointly as \(x\) and \(z,\) and \(y=\frac{3}{2}\) when \(x=2\) and \(z=10\).

Use a graphing calculator to check Example \(5 .\) Perform the check using $$ \begin{aligned} &y_{1}=\left(9 x^{2}+x^{3}-5\right) /\left(x^{2}-1\right)\\\ &y_{2}=x+9+(x+4) /\left(x^{2}-1\right), \text { and } y_{3}=y_{2}-y_{1} \end{aligned} $$

Simplify. $$ \frac{m^{2}-t^{2}}{m^{2}+t^{2}+m+t+2 m t} $$

The tension \(T\) on a string in a musical instrument varies jointly as the string's mass per unit length \(m,\) the square of its length \(l,\) and the square of its fundamental frequency \(f . \mathrm{A} 2-\mathrm{m}\) long string of mass \(5 \mathrm{gm} / \mathrm{m}\) with a fundamental frequency of 80 has a tension of \(100 \mathrm{N}(\) Newtons). How long should the same string be if its tension is going to be changed to \(72 \mathrm{N} ?\)

Find the variation constant and an equation of variation if y varies directly as \(x\) and the following conditions apply. \(y=0.9\) when \(x=0.5\)
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