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Chapter 7: Problem 51

Solve each formula for \(k\) $$ y=\frac{k}{x} $$

Short Answer

Expert verified
The solution for \(k \) is \(k = y \cdot x \)

Step by step solution

01

- Isolate k

To solve the formula for \(k \), start by isolating \(k \). Multiply both sides of the equation by \(x \) to eliminate the fraction. This gives you: \[ y \cdot x = k \]
02

- Simplify

Now, simplify the expression to solve for \(k \): \[ k = y \cdot x \]

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation is a fundamental skill in solving equations. It involves rearranging and simplifying expressions to isolate the variable you're solving for. This often includes adding, subtracting, multiplying, or dividing both sides of an equation by the same quantity. Manipulating equations allows us to find unknown values and understand the relationships between different variables.
In our example, the original equation is given as
\[ y = \frac{k}{x} \]
To solve for \(k\), you need to manipulate the equation algebraically. This ensures we're following mathematical principles while making the process simpler and more straightforward. Understanding how to manipulate expressions effectively will make solving complex problems more manageable.
Isolating Variables
Isolating variables is the method of rearranging an equation so that the variable of interest stands alone on one side of the equation. This makes it easier to understand what the variable represents and to solve for it.
In the provided exercise, the goal is to isolate \(k\). This is achieved by eliminating the fraction that involves \(k\). By multiplying both sides of the equation by \(x\), we remove \(x\) from the denominator:
\[ y \times x = k \]
Here, you directly isolate \(k\). This technique can be applied to more complex equations, helping you to systematically solve for any variable by isolating it on one side of the equation.
Fraction Elimination
Fractions can complicate equations, but eliminating them can simplify the problem significantly. To eliminate a fraction, you can multiply every term in the equation by the denominator.
In the exercise, starting with the equation:
\[ y = \frac{k}{x} \]
the denominator is \(x\). By multiplying both sides of the equation by \(x\), you get:
\[ y \times x = k \]
This process effectively removes the denominator, transforming a complex fraction into a simpler multiplication. This makes the variable \(k\) easier to work with and solve for. Understanding fraction elimination is crucial for tackling a wide variety of algebraic problems confidently.

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Most popular questions from this chapter

Use personal experience or intuition to determine whether the situation suggests direct or inverse variation. The number of days from now until December 25 and the magnitude of the frenzy of Christmas shopping

Connie McNair's boat goes \(12 \mathrm{mph}\). Find the rate of the current of the river if she can go \(6 \mathrm{mi}\) upstream in the same amount of time she can go \(10 \mathrm{mi}\) downstream.

Solve each problem involving direct or inverse variation. If \(z\) varies directly as \(x,\) and \(z=30\) when \(x=8,\) find \(z\) when \(x=4\)

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{r^{2}+r s-12 s^{2}}{r^{2}-r s-20 s^{2}} \div \frac{r^{2}-2 r s-3 s^{2}}{r^{2}+r s-30 s^{2}}$$

Solve each formula for \(k\) $$ y=\frac{k}{x^{2}} $$
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