Problem 10 - Algebra 1

Question

When \(x=4, y=6 .\) For the given type of variation, find an equation that relates \(x\) and \(y .\) Then find the value of \(y\) when \(x=8\). \(x\) and \(y\) vary directly.

Step-by-Step Solution

Answer

The equation that relates \(x\) and \(y\) is \(y=1.5x\). The value of \(y\) when \(x=8\) is \(y=12\)

Step 1: Determine the Constant of Variation

In direct variation, the constant of variation \(k\) is given by \(k=\frac{y}{x}\). Here, for \(x=4\) and \(y=6\), we calculate \(k=\frac{6}{4}=1.5\)

Step 2: Formulate the Equation of Variation

Substitute the calculated value of \(k\) into the formula for direct variation \(y=kx\). This gives us the equation \(y=1.5x\)

Step 3: Find the Value of y when x=8

Substitute \(x=8\) into the formulated equation \(y=1.5x\). This gives us \(y=1.5*8=12\)

Key Concepts

Constant of Variation

The constant of variation is a crucial element in understanding direct variation. When two variables, such as \(x\) and \(y\), vary directly, it means they change together in the same proportion.Here's how to find that proportion, called the constant of variation.
To find the constant of variation, represented by \(k\), use the formula \(k = \frac{y}{x}\).
This formula asks you to divide \(y\) by \(x\).This ratio remains consistent whenever \(x\) and \(y\) change, as long as they stay in direct variation.
In the example, with \(x = 4\) and \(y = 6\), the constant of variation comes out to \(1.5\).This means that for every single unit increase in \(x\), \(y\) increases by \(1.5\) units.This constant ties \(x\) and \(y\) together in their direct relationship.

Formulate Equation

After determining the constant of variation, the next step is to formulate the equation of variation.This equation uses the constant \(k\) to express the direct relationship between \(x\) and \(y\).
In direct variation, the relationship can be expressed with the equation \(y = kx\).\(k\) is already known from calculating the previous step.
Using the provided example, we substitute \(k = 1.5\) into \(y = kx\), resulting in the equation \(y = 1.5x\).This equation clearly shows how \(x\) and \(y\) relate: \(y\) is always \(1.5\) times \(x\).
Now, you have a formula that explains the direct variation between \(x\) and \(y\). You can use this formula to find \(y\) for any value of \(x\), as long as the conditions of the problem remain unchanged.

Substitution

Once you have the equation of variation, substitution allows you to find unknown values.In this context, substitution means replacing the given variable with its specific value.
Let's follow the original problem: you want to find \(y\) when \(x = 8\) using the equation \(y = 1.5x\).
Substitute \(8\) for \(x\) in the equation:

• Replace \(x\) with \(8\), giving \(y = 1.5 \times 8\).
• Perform the multiplication: \(1.5 \times 8 = 12\).

Therefore, \(y = 12\) when \(x = 8\).
Substitution helps to find specific outcomes based on the general relationship established by your equation.This technique allows you to solve for \(y\) given any specific value of \(x\) and vice versa.