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Chapter 7: Problem 50

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

Short Answer

Expert verified
The solution for \(k\) is \( k = \frac{y}{x^{2}} \).

Step by step solution

01

Understand the problem

The given equation is a quadratic equation: - \(y = k x^{2}\). The objective is to solve for the variable \(k\).
02

Isolate the variable

To isolate \(k\), we need to get \(k\) alone on one side of the equation. In this case, divide both sides of the equation by \(x^{2}\): \( \frac{y}{x^{2}} = k \).
03

Express the solution

After the division, we obtain the expression for \(k\): \( k = \frac{y}{x^{2}} \). This is the solution to the equation.

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Key Concepts

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

Quadratic Equation
In mathematics, a quadratic equation is one that involves the square of a variable, typically written in the form \[ ax^2 + bx + c = 0 \]. For this particular exercise, the equation is simpler: \[ y = kx^2 \]. Here, it involves only one term with the variable raised to the power of 2. The goal is to solve for the constant 'k'. Understanding the form and structure of a quadratic equation can be crucial when manipulating it for various purposes.
A quadratic equation typically features:
  • One squared term (\(x^2\)).
  • No linear (\(x\)) or constant (\(c\)) terms in some cases.
It's straightforward in this exercise, involving fewer steps for isolating the variable 'k'.
Isolating Variables
When solving equations, one fundamental skill is isolating the variable you want to solve for. This means rearranging the equation so that the variable is alone on one side. In this example, we need to isolate 'k' in\[ y = kx^2 \].
To isolate 'k', perform the following steps:
  • Identify the term involving 'k' (here, it is \(kx^2\)).
  • Divide both sides of the equation by the coefficient of 'k', which is \(x^2\)
This gives us \( \frac{y}{x^2} = k \), effectively isolating 'k' on the right-hand side. Always remember that whatever operation you perform on one side of the equation, you must also perform on the other side to maintain equality.
Formula Manipulation
Manipulating formulas is a powerful tool in algebra, used to solve for a particular variable or to simplify expressions. In this exercise, we started with the formula \[ y = kx^2 \]. To solve for 'k', we needed to manipulate this formula.
The steps to manipulate this formula included:
  • Dividing both sides by \(x^2\).
  • Ensuring the operations applied maintain the balance in the equation.
The resulting formula \[ k = \frac{y}{x^2} \] can be used to find 'k' for any given values of 'y' and 'x'. Knowing how to properly manipulate formulas can help you not only solve equations but also understand relationships between different variables more deeply.

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Most popular questions from this chapter

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{6 s^{2}+17 s+10}{s^{2}-4} \cdot \frac{s^{2}-2 s}{6 s^{2}+29 s+20}$$

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{7}{5}-\frac{3}{4} $$

Use personal experience or intuition to determine whether the situation suggests direct or inverse variation. The number of candy bars you buy and your total price for the candy

Rewrite each rational expression with the indicated denominator. $$ \frac{m-4}{6 m^{2}+7 m-3}=\frac{?}{12 m^{3}+14 m^{2}-6 m} $$

Solve each problem involving direct or inverse variation. If \(m\) varies inversely as \(r,\) and \(m=12\) when \(r=8,\) find \(m\) when \(r=16\)
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