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Chapter 7: Problem 89

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

Short Answer

Expert verified
k = 3

Step by step solution

01

- Substitute given values

Use the values given in the problem. Substitute \( y = 1 \) and \( x = 3 \) into the equation \( y = \frac{k}{x} \). This gives us: \[ 1 = \frac{k}{3} \]
02

- Solve for \( k \)

To isolate \( k \), multiply both sides of the equation by 3: \[ 1 \cdot 3 = \frac{k}{3} \cdot 3 \]This simplifies to: \[ k = 3 \]

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Key Concepts

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

Substitution
Substitution is a fundamental concept in algebra. It involves replacing variables in an equation with given values, making the equation easier to solve. For instance, in the exercise, we start with the equation: \[ y = \frac{k}{x} \]We are given specific values for \(y\) and \(x\): \(y = 1\) and \(x = 3\). By substituting these values into the equation, we replace \(y\) with 1 and \(x\) with 3: \[ 1 = \frac{k}{3} \]This process simplifies the equation and prepares it for the next steps, making it easier to solve for the unknown variable.
Solving for Variables
In algebra, solving for variables means isolating a particular variable to find its value. Once we have substituted the given values into the equation \[ 1 = \frac{k}{3} \]we need to solve for \(k\). To do this, we want \(k\) alone on one side of the equation. Currently, \(k\) is divided by 3. To isolate \(k\), we perform the opposite operation 鈥� multiplication.By multiplying both sides of the equation by 3, we get: \[ 1 \cdot 3 = \frac{k}{3} \cdot 3 \]This simplifies to: \[ 3 = k \]Now we've successfully solved for the variable \(k\), finding its value to be 3.
Multiplication
Multiplication is one of the basic arithmetic operations and is essential in solving algebraic equations. When an equation involves a fraction, as in our exercise \[ 1 = \frac{k}{3} \]to remove the fraction, we use multiplication. The equation tells us that \(k\) divided by 3 equals 1. To cancel out the division, we multiply both sides by 3: \[ 1 \cdot 3 = \frac{k}{3} \cdot 3 \]Multiplying 1 by 3 gives us 3, and multiplying \(\frac{k}{3}\) by 3 cancels out the 3 in the denominator, leaving us with \(k\). This step illustrates how multiplication can help isolate variables and simplify equations, leading to the solution: \[ k = 3 \]

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