Problem 4 - College Algebra with Corequisite Support

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as \(x\) and when \(x=6, y=12\).

Step-by-Step Solution

Answer

The equation is \( y = 2x \).

Step 1: Understand Direct Variation

When we say that \( y \) varies directly as \( x \), it means that \( y = kx \), where \( k \) is a constant of proportionality. We need to find \( k \) using the given condition.

Step 2: Substitute Known Values to Find k

We know that when \( x = 6 \), \( y = 12 \). Substitute these values into the direct variation equation \( y = kx \): \( 12 = k \times 6 \).

Step 3: Solve for the Constant k

Solve the equation \( 12 = 6k \) for \( k \). Divide both sides by 6 to get \( k = \frac{12}{6} = 2 \).

Step 4: Write the Final Equation

Now that we have \( k = 2 \), substitute back into the direct variation equation to get \( y = 2x \). This is the equation that describes the relationship between \( y \) and \( x \).

Key Concepts

Constant of Proportionality

In mathematics, when two variables vary directly, they are connected by a special number called the constant of proportionality. This constant 'k' helps quantify how one variable changes with another. It acts as a multiplier that scales the change in one variable to match the change in the other.

Direct variation can be expressed in the form of the equation:

• \( y = kx \)

Here:

• 'y' and 'x' are variables of interest.
• 'k' is the constant of proportionality.

When you know the value of 'k', you can predict the behavior of 'y' for any given 'x', and vice versa. In our example, substituting the values from the original exercise, where when \( x=6 \), \( y=12 \), we find \( k = 2 \). This means for every increase of 1 unit in 'x', 'y' increases by 2 units.

Algebraic Equation

An algebraic equation is a mathematical statement that shows the equality of two expressions. It often involves variables and constants connected by operations like addition, subtraction, multiplication, or division.

In the context of direct variation, we use algebraic equations to define the relationship between the variables. For direct variation, the equation takes the form:

• \( y = kx \)

This equation is fundamental in illustrating how two quantities change together. Each component of the equation has significance:- 'y' represents the dependent variable.- 'x' is the independent variable.- 'k' is the constant of proportionality.Through this equation, we can understand how 'y' responds to changes in 'x', given 'k'. Using our solution, the equation becomes \( y = 2x \), reflecting how 'y' is scaled by 2 for every value of 'x'.

Solving Equations

Solving equations is the process of finding unknown values that satisfy the equation's conditions. In many algebraic contexts, such as our problem, it involves isolating the variable or constant of interest.

When given specific values of variables, we can solve for the constant of proportionality by substituting them into the direct variation equation. Given the equation \( 12 = 6k \) from substituting \( x = 6 \) and \( y = 12 \), solving it involves:

• Dividing both sides by 6: \( \frac{12}{6} = k \)
• Resulting in \( k = 2 \)

Once 'k' is known, we can rewrite the original formula to predict how variables interact. Solving such equations can initially feel challenging, but like a puzzle, it starts to make more sense as you practice. Understanding the process ensures you can efficiently handle similar problems in various applications.