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Chapter 7: Problem 90

Find \(k,\) given that \(y=1\) and \(x=3 .\) See Sections 2.3 and 2.5. $$ y=\frac{k}{x^{2}} $$

Short Answer

Expert verified
k = 9

Step by step solution

01

Identify the Given Values

Identify the values given in the problem: \(y = 1\) and \(x = 3\).
02

Substitute the Values

Substitute the given values into the equation: \(y = \frac{k}{x^2}\). This gives us: \(1 = \frac{k}{3^2}\).
03

Simplify the Denominator

Simplify the denominator: \(3^2 = 9\). So the equation becomes: \(1 = \frac{k}{9}\).
04

Solve for \(k\)

Solve for \(k\) by multiplying both sides of the equation by 9: \(k = 1 \times 9\). Therefore, \(k = 9\).

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Key Concepts

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

substitution method
The substitution method is a powerful tool in algebra. It involves replacing one variable with another value to simplify an equation.
For instance, in our problem, we are given: \[ y = \frac{k}{x^2} \] and it states that when \( y = 1\) and \( x = 3\).
Here鈥檚 why substitution is useful:
  • It allows us to bring values we know into an equation, making it easier to solve.
  • Once we substitute the values into the equation, we deal with numbers rather than symbols, which can be less intimidating.
In the exercise, substituting \( y = 1\) and \( x = 3\) gives us: \[ 1 = \frac{k}{3^2} \]. By substituting correctly, we can break down the problem step by step.
simplifying expressions
Simplifying expressions is crucial before solving. It means reducing them to their simplest form, making calculations easier.
In the given problem, after substituting the values, we have: \[ 1 = \frac{k}{3^2} \].
We simplify the denominator next. The original expression has \( 3^2\), which simplifies to 9. Now our equation looks like:
\[ 1 = \frac{k}{9} \].
Key points when simplifying expressions include:
  • Always follow the order of operations, also known as BIDMAS/BODMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction).
  • Look for opportunities to reduce fractions or combine like terms for ease.
Once simplified, you鈥檙e often a step away from the solution.
algebra basics
Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. Here are some fundamentals:
  • Variables represent numbers and can stand in for unknowns. For example, \(k\) and \(x\) in our problem.
  • Equations are mathematical statements indicating that two expressions are equal.
Now, let鈥檚 focus on solving for the variable \(k\). This final step involves isolating \(k\) on one side of the equation:
Based on the simplified equation:
\[ 1 = \frac{k}{9} \]
To solve for \( k\), we multiply both sides by 9: \[ 1 \times 9 = k \].
This simplifies to \ k = 9 \. Understanding these basics ensures you can solve algebra problems efficiently and effectively.

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