Problem 25 - Intermediate Algebra

Question

If \(y\) is directly proportional to \(x\), and \(y=36\) when \(x=48\), find the value of \(y\) when \(x=12\).

Step-by-Step Solution

Answer

When \( x = 12 \), \( y = 9 \).

Step 1: Define Direct Proportionality

Since \( y \) is directly proportional to \( x \), we can express this relationship as \( y = kx \), where \( k \) is the constant of proportionality.

Step 2: Find Constant of Proportionality

We know that \( y = 36 \) when \( x = 48 \). Substitute these values into the equation to find \( k \): \[ 36 = k imes 48 \]. To solve for \( k \), divide both sides by 48: \[ k = \frac{36}{48} = \frac{3}{4} \].

Step 3: Use the Constant to Find New \( y \)

Now that we have \( k = \frac{3}{4} \), apply this to find \( y \) when \( x = 12 \). Substitute \( x = 12 \) into the formula \( y = kx \): \[ y = \frac{3}{4} imes 12 \].

Step 4: Calculate the Value of \( y \)

Carry out the multiplication: \[ y = \frac{3}{4} imes 12 = 9 \]. Thus, when \( x = 12 \), \( y = 9 \).

Key Concepts

Algebraic Equations

An algebraic equation is a mathematical statement where two expressions are set equal to one another. For example, in the context of direct proportionality, we often use the equation \( y = kx \). Here, \( y \) and \( x \) are variables, while \( k \) is a constant. This equation illustrates a basic relationship where the value of \( y \) changes in accordance with \( x \).
In algebraic equations, understanding the relationship between variables is key. The variables can be manipulated by substituting known values into the equation to find unknown values. In our exercise, the given relation \( y = kx \) helps us solve for one variable if the other and the constant are known. The steps involve simple calculations, like multiplying or dividing, which makes them essential tools in algebra.

Constant of Proportionality

The constant of proportionality \( k \) is a special number that helps maintain the relationship between two variables. If two quantities are directly proportional, like \( y \) and \( x \) in our exercise, you can express their relationship as \( y = kx \).
Finding \( k \) is crucial because it allows you to predict how changes in one variable affect the other. In the example given, we used the values \( y = 36 \) and \( x = 48 \) to determine that \( k = \frac{3}{4} \).
After obtaining \( k \), we can use it to find the value of \( y \) for different values of \( x \). This operation underlines the constancy of \( k \) across different scenarios, demonstrating that \( k \) governs the direct relationship between \( y \) and \( x \). Recognizing and calculating \( k \) is foundational to interpreting linear relationships.

Linear Relationships

Linear relationships describe a straight-line connection between two variables. They are characterized by equations of the form \( y = kx + c \); however, when it comes to direct proportionality, the form simplifies to \( y = kx \), implying that \( c \) is zero.
This simplicity means that the graph of the relationship will always pass through the origin, reflecting that when one variable is zero, so is the other.
In our exercise, the direct proportionality illustrated this by consistently using the same \( k \) for different values of \( x \) and \( y \).
Understanding linear relationships is fundamental in algebra because they display an unchanging rate of change. This predictability makes linear models a powerful tool for analyzing real-world data, where direct relationships between factors are common.