Problem 1 - College Algebra

Question

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies directly as \(x . y=65\) when \(x=5 .\) Find \(y\) when \(x=12\)

Step-by-Step Solution

Answer

The value of \(y\) when \(x = 12\) is 156.

Step 1: Finding the constant of variation

We know from the given information that \(y = kx\). We also know that \(y=65\) when \(x=5\). Substituting these values into the equation, we can solve for \(k\): \(65 = k(5)\). Simplifying this, we get \(k = 65/5 = 13.\)

Step 2: Finding the missing value using the constant

With the found value of \(k\), we can find \(y\) when \(x=12\). We substitute these values into the variation equation \(y = kx\) and get \(y = 13 * 12 = 156.\)

Key Concepts

Constant of Variation

When we talk about the constant of variation, we are referring to a fixed number that relates two variables that are directly proportional to each other in a variation problem. It is represented by the symbol 'k' in algebraic equations. In the given exercise, we determine this constant by using the information that when the variable x is 5, the variable y is 65.

To find the constant of variation, we use the equation y = kx and plug in the known values, resulting in the equation 65 = k(5). By solving for k, we divide 65 by 5, which gives us k = 13. This value of k remains unchanged throughout the problem and is crucial for finding y for other values of x.

Direct Variation

The concept of direct variation describes a linear relationship between two variables in which one variable is a constant multiple of the other. This means as one variable increases, the other variable increases at a consistent rate, and vice versa.

Direct variation is expressed algebraically as y = kx where y and x are the variables and k is the non-zero constant of variation. This relationship is important in understanding how changes in one variable will affect another. For example, returning to our exercise, once we know k is 13, we can predict y for any value of x. If x grows, so does y, and it does so at a rate 13 times that of x. When x is 12, y is calculated to be 156, which perfectly illustrates the concept of direct variation.

Algebraic Equations

In mathematics, algebraic equations are a fundamental tool used to solve problems involving unknown variables. These equations consist of expressions set equal to each other, containing constant numbers, variables (like x and y), and sometimes coefficients (like our constant of variation k).

Algebraic equations can be as simple as linear equations with one variable, or as complex as nonlinear equations with multiple variables. The process of solving an equation involves finding the value(s) of the variable(s) that make the equation true. In our exercise, the algebraic equation y = 13x is derived from the original direct variation equation after finding the constant of variation. By substituting x with 12, we solve the algebraic equation to find that y equals 156, reaffirming the relationship initially described between y and x.