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Chapter 7: Problem 49

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

Short Answer

Expert verified
k = y / x.

Step by step solution

01

Understand the Equation

The given equation is the linear equation y = k x. We need to solve this for the variable k.
02

Isolate the Variable k

To isolate k, we need to get k by itself on one side of the equation. To do this, divide both sides of the equation by x. This yields:y / x = k x / x.
03

Simplify the Equation

Now simplify the right side of the equation:y / x = k. Thus, we have isolated k.

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Key Concepts

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

linear equations
A linear equation is a simple, yet fundamental type of algebraic expression. It can be written in the form of ax + b = 0 or similar variations. In the specific exercise, the linear equation is given as y = kx.
Linear equations have the following characteristics:
  • They graph as straight lines on a Cartesian plane.
  • They have a constant rate of change.
Understanding linear equations is crucial for solving for unknown variables and analyzing relationships between them.
isolating variables
Isolating a variable means manipulating an equation to get the variable alone on one side. In the exercise, we aim to solve for the variable k.
To isolate k, you should follow these steps:
  • Identify the variable you need to isolate.
  • Perform algebraic operations to remove other terms or coefficients.
  • Ensure that you perform the same operations on both sides of the equation to keep it balanced.
For instance, in the given equation y = kx, we need to isolate k by dividing both sides by x.
algebraic manipulation
Algebraic manipulation involves performing operations to rearrange and simplify equations. These are the tools you'll often use:
  • Addition/Subtraction: Used to move terms from one side of an equation to the other.
  • Multiplication/Division: Used to eliminate coefficients or to isolate variables.
  • Distribution: Involves multiplying through parentheses to simplify expressions.
In the given problem, we use division to simplify and isolate the variable k. Mastering these manipulations is key to solving various types of equations.
division in equations
Division is used to solve for a variable when that variable is multiplied by another term. For the equation y = kx, we divide both sides by x to solve for k.
The steps are:
  • Identify the multiplication relationship involving the variable you need to isolate.
  • Divide both sides of the equation by the same non-zero term to maintain equality.
  • Simplify the resulting equation.
So, in this problem, we perform the division y/x = k, which isolates k effectively. Understanding when and how to use division is vital for manipulating equations correctly.

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Most popular questions from this chapter

Fill in each blank with the correct response. (a) If the constant of variation is positive and \(y\) varies directly as \(x,\) then as \(x\) increases, \(y=\) ____. (increases/decreases) (b) If the constant of variation is positive and \(y\) varies inversely as \(x,\) then as \(x\) increases, \(V\) ____. (increases/decreases)

Solve each variation problem. If the temperature is constant, the pressure of a gas in a container varies inversely as the volume of the container. If the pressure is 10 lb per \(\mathrm{ft}^{2}\) in a container with volume \(3 \mathrm{ft}^{3}\), what is the pressure in a container with volume \(1.5 \mathrm{ft}^{3} ?\)

Solve each problem involving direct or inverse variation. If \(z\) varies inversely as \(x,\) and \(z=50\) when \(x=2,\) find \(z\) when \(x=25\)

Determine whether each equation represents direct or inverse variation. $$ y=200 x $$

Solve each problem. In a certain fraction, the denominator is 6 more than the numerator. If 3 is added to both the numerator and the denominator, the resulting fraction is equivalent to \(\frac{5}{7} .\) What was the original fraction (not written in lowest terms)?
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