Problem 65 - Algebra 2

Question

Solve each equation or formula for the specified variable. $$ 4 x^{2}-5 x y+2=3, \text { for } y $$

Step-by-Step Solution

Answer

The solution for \( y \) is \( y = \frac{4x^2 - 1}{5x} \).

Step 1: Set up the equation

We begin with the original equation: \(4x^2 - 5xy + 2 = 3\). Our goal is to solve for \(y\). First, we need to isolate terms involving \(y\) on one side of the equation.

Step 2: Subtract 2 from both sides

To isolate terms with \(y\), subtract 2 from both sides of the equation: \(4x^2 - 5xy + 2 - 2 = 3 - 2\). This simplifies to \(4x^2 - 5xy = 1\).

Step 3: Solve for y

Now, we solve for \(y\) by isolating it. Since \(y\) is multiplied by \(-5x\), divide the entire equation by \(-5x\): \(y = \frac{4x^2 - 1}{5x}\).

Key Concepts

Isolation of Variables

Isolation of variables is a fundamental technique in solving equations. It involves rearranging an equation so that the targeted variable appears by itself on one side. This process helps us find the value of that variable in terms of other known quantities or numbers.

• Consider the equation given in the exercise: \(4x^{2} - 5xy + 2 = 3\). Here, we want to solve for \(y\), which means we need to have \(y\) alone on one side of the equation.
• To start, subtract 2 from both sides of the equation to keep the \(y\) term alone: \(4x^{2} - 5xy = 1\).
• Next, since -5x is multiplied with \(y\), divide both sides by \(-5x\) to isolate \(y\): \(y = \frac{4x^{2} - 1}{5x}\).

This illustrates how isolating means getting the variable by itself, often through a combination of addition, subtraction, multiplication, or division. It is crucial for solving linear and other equations.

Quadratic Equations

Quadratic equations involve a variable raised to the second power, typically in the form \(ax^{2} + bx + c = 0\). They can be recognized by the \(x^{2}\) term, which governs the structure of the equation.

• The presence of \(4x^{2}\) in the original equation shows that it includes a quadratic element crucial in understanding its complexity.
• Quadratic equations can present solutions that are real or complex numbers, and there are different methods to solve them including factoring, using the quadratic formula, or completing the square.

In the provided exercise, the term \(4x^{2}\) is part of a larger equation we are trying to rearrange to isolate \(y\). Remember, while \(y\) is not squared, the equation involves handling the quadratic term gracefully to reach isolation.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations to simplify them or to isolate variables. It involves applying mathematical operations such as addition, subtraction, multiplication, division, and the distributive property.

• In our exercise, two main algebraic manipulations were performed: subtracting numbers from both sides and dividing the whole equation by a non-zero quantity (\(-5x\)).
• Subtracting elements from both sides is typically one of the first steps to simplify equations, as seen when subtracting 2 to transform \(4x^{2} - 5xy + 2 = 3\) into \(4x^{2} - 5xy = 1\).
• Then, by dividing the whole equation by \(-5x\), we successfully isolated \(y\) in terms of \(x\).

Algebraic manipulation is fundamental in solving equations, allowing us to rearrange and simplify until the solution appears evident. It's all about being methodical and careful with every step to ensure accuracy.