Problem 20 - College Algebra

Question

Write an equation that expresses each relationship. Then solve the equation for y. \(x\) varies directly as \(z\) and inversely as the sum of \(y\) and \(w\)

Step-by-Step Solution

Answer

The equation expressing the direct and inverse relationship for \(y\) is \(y = \frac {1}{z} - w\), given that x varies directly as z and inversely as the sum of \(y\) and \(w\).

Step 1: Writing down the equation for direct variation

The statement 'x varies directly as z' tells us that \(x = kz\), where \(k\) is the constant of variation.

Step 2: Translating the statement involving indirect variation into an equation

The statement 'x varies inversely as the sum of y and w' can be translated into the following equation: \(x = \frac{k}{y+w}\), where \(k\) is the constant of variation.

Step 3: Combining the equations

Since both of the equations represent the value of \(x\), they can be set equal to each other: \(kz = \frac{k}{y+w}\)

Step 4: Solve for y

To solve for \(y\), first, cancel out \(k\) from both sides, then cross-multiply and rearrange the equation: \((y+w)z =1, y = \frac {1}{z} - w\)

Key Concepts

Solving Equations

When faced with the task of solving equations, it's important to understand that you're looking for the value or values that make the equation true. The process is like a puzzle where you must decode the clues given to find the missing number.

In our exercise, the clue comes in the form of direct and inverse variation. You're given two relationships involving variables x, y, z, and w. The trick is to express these relationships as algebraic equations and then manipulate those equations to isolate the variable y.

This process may often involve multiple steps, such as rearranging terms, combining like terms, or applying properties of equality. It's key to do these steps in a systematic way to avoid any confusion. Substituting values, when known, can also be a part of solving the equation and checking if the values satisfy the original equation is always a good final step to validate the solution.

Constant of Variation

The constant of variation, denoted as k in our exercise, is a fixed number that relates two variables which are directly proportional or inversely proportional to each other. In direct variation, as one variable increases, the other also increases by a factor of k.

On the other hand, in inverse variation, one variable increases as the other decreases, but the product or quotient (depending on the type of variation) is still equal to the same constant k.

In the given problem, we see that x varies directly as z and inversely as y + w. This means that, as z gets larger, x will also get larger by multiplying z by k for direct variation. Conversely, as the sum of y and w gets larger, x gets smaller since x is equal to k divided by y + w for inverse variation.

Algebraic Expressions

Understanding algebraic expressions is crucial in representing and solving mathematical problems involving variations. An algebraic expression is a combination of constants, variables, and arithmetic operations (like addition, subtraction, multiplication, and division).

In our case, the expressions for direct and inverse variations are set up to reflect the proportional relationships between the variables. By setting the direct and inverse variation expressions equal to each other, we create a single equation that allows us to isolate and solve for the variable of interest, y.

The essence of algebra is to manipulate these expressions through a series of algebraic steps, preserving the equality and solving for variables. Additionally, understanding how to combine expressions, factor, and expand them can help to simplify and solve even the most complex equations.