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Chapter 6: Problem 79

Solve formula for the specified variable. \(m=\frac{k F}{a}\) for \(F\)

Short Answer

Expert verified
F = \frac{m \times a}{k}

Step by step solution

01

- Identify the given formula

The formula provided is \(m = \frac{k F}{a}\). The goal is to solve this formula for the variable \(F\).
02

- Isolate the fraction

First, multiply both sides of the equation by \(a\) in order to eliminate the denominator. This gives: \(m \times a = k F\).
03

- Solve for F

Next, divide both sides of the equation by \(k\) to isolate \(F\). This results in: \(F = \frac{m \times a}{k}\).

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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 fundamental skill in algebra. It's the process of rearranging an equation so that one specific variable stands alone on one side of the equation. Here, our goal is to solve for the variable F in the given formula: \(m = \frac{k F}{a}\). To isolate F, we must remove the other elements (k and a) from its side. This usually involves reversing operations. For example:
  • If a variable is multiplied by another term, you divide both sides by that term.
  • If a variable is divided by a term, you multiply both sides by that term.
In our case, we needed to get rid of the fraction first by multiplying both sides by a. Next, we divided both sides by k to have F by itself. Understanding the goal of isolating variables is crucial, as it can be applied to many different types of equations.
algebraic manipulation
Algebraic manipulation involves rearranging and simplifying algebraic expressions and equations using a variety of operations, such as addition, subtraction, multiplication, and division. Here's a step-by-step breakdown using our example formula \(m = \frac{k F}{a}\):
  • Firstly, identify the operations you need to reverse. To deal with the fraction, you must multiply both sides by the denominator a.
  • This step eliminates the fraction and simplifies the equation to: \(m \times a = k F\).
  • Next, isolate the variable F by dividing both sides by k: \(F = \frac{m \times a}{k}\).
It's all about understanding how each operation affects the terms and reversing them systematically. Mastering algebraic manipulation makes solving complex equations more manageable.
fractions in equations
Dealing with fractions in equations can seem tricky at first, but once you grasp a few key principles, it becomes much simpler. The main challenge is often the presence of a fraction that can complicate an equation. To handle fractions:
  • Always aim to eliminate the fraction early on.
  • Multiply both sides of the equation by the denominator to clear the fraction.
In our specific formula \(m = \frac{k F}{a}\), we observed this process by multiplying both sides by a, yielding \(m \times a = k F\). This step cleared the fraction, making it easier to isolate the variable F. Understanding how to treat fractions in algebraic equations is essential to working efficiently through these types of problems.

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Most popular questions from this chapter

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{x^{3}-27}{x-3} $$

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{k m+4 k-4 m-16}{k m+4 k+5 m+20} $$

Multiply. Write each answer in lowest terms. $$ \frac{12 x^{4}}{18 x^{3}} \cdot \frac{-8 x^{5}}{4 x^{2}} $$

Multiply. Write each answer in lowest terms. $$ \frac{(t-2)^{2}}{4 t^{2}} \cdot \frac{2 t}{t-2} $$

Multiply or divide. Write each answer in lowest terms. $$ \frac{(x+4)^{3}(x-3)}{x^{2}-9} \div \frac{x^{2}+8 x+16}{x^{2}+6 x+9} $$
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