Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ 4 x^{3}+7 x y^{2}=2 y^{3} $$

Step-by-Step Solution

Answer

Exact values can't be determined with one equation; more info needed.

Step 1: Solving for one variable

Since we have a single equation with two variables, let's solve for one variable in terms of the other. Let's express \(y\) in terms of \(x\). Rearranging the terms in the equation, we get: \[ 4x^3 = 2y^3 - 7xy^2 \].Now, try to solve for \(y\) if possible.

Step 2: Factor and simplify expressions

Notice that \(y\) appears in both terms on the right. We can try to factor it to simplify:\[ 4x^3 = y^2(2y - 7x) \].From here, solving for \(y\) straightforwardly is difficult without additional information. Additional constraints or a secondary equation might be necessary.

Step 3: Conclusion on the form of expression

With the current equation form, finding a simple expression for \(y\) in terms of \(x\) is challenging without assumptions or additional equations. Each value of \(x\) could potentially lead to different solutions for \(y\), or vice versa.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a crucial skill in solving equations. It involves rearranging equations and expressions using various algebraic operations, such as addition, subtraction, multiplication, division, and factoring. In our example, the chief aim is to express one variable in terms of another, which often involves moving terms around the equation or factoring them to simplify the expression.

In the original exercise, we start with the equation:

• \[ 4x^3 + 7xy^2 = 2y^3 \]

We manipulate it by moving the terms involving \( y \) to one side. This rearranges the equation to:

• \[ 4x^3 = 2y^3 - 7xy^2 \]

Further, recognizing that \( y \) is common in the right-hand terms, factoring helps simplify it into:

• \[ 4x^3 = y^2(2y - 7x) \]

This process of moving terms and factoring is a key part of algebraic manipulation, helping transform complex equations into more manageable forms.

System of Equations

A system of equations involves multiple equations that are solved together to find a common solution. Each equation expresses a relationship between the variables in play. Systems aren’t always linear; they can include more complex, non-linear equations as seen in this exercise.

In the given problem, technically, we have only one equation:

• \[ 4x^3 + 7xy^2 = 2y^3 \]

However, solving for one variable, say \( y \), in terms of \( x \) can resemble solving a system. Ideally, another equation specifying a relationship between \( x \) and \( y \) would provide a system of equations. This additional equation would help pinpoint specific solutions.

While this exercise doesn't provide a second equation, understanding how systems work can guide problem-solving. Multiple equations can offer more constraints, leading to a unique solution or understanding multiple solutions' nature.

Non-linear Equations

Non-linear equations include terms that are not purely linear (i.e., not proportional to their variables to the first power). In this exercise, we encounter a non-linear equation with terms like \( x^3 \) and \( y^3 \). Such equations can be more complex to solve due to their curvy graph representations, which don't form straight lines.

The given equation:

• \[ 4x^3 + 7xy^2 = 2y^3 \]

is non-linear because of the cubic and mixed product terms involving \( x \) and \( y \). Solving non-linear equations often requires special techniques, such as factoring (as attempted here), graphing, or using numerical methods for approximation in more intricate cases.

It's essential to understand that without simplifying assumptions or additional information, non-linear equations might not yield unique solutions. Different values for one variable could lead to several possible values for another. This complexity is why some non-linear systems require more equations to confine the range of solutions accurately.