Problem 6 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the given equation. $$ 2-3 y=8 $$

Step-by-Step Solution

Answer

The solution to the equation \(2-3y=8\) is \(y=-2\).

Step 1: Add 3y to both sides of the equation.

To isolate y, we will first add 3y to both sides of the equation: \(2 - 3y + 3y = 8 + 3y\) This simplifies to: \(2 = 8 + 3y\)

Step 2: Subtract 2 from both sides of the equation.

Next, we will subtract 2 from both sides of the equation: \(2 - 2 = 8 + 3y - 2\) This simplifies to: \(0 = 6 + 3y\)

Step 3: Divide both sides by 3.

Finally, we will divide both sides of the equation by 3 to solve for y: \(\frac{0}{3} = \frac{6 + 3y}{3}\) This simplifies to: \(0 = 2 + y\) Now, we can easily see that y = -2 is the solution to the equation.

Key Concepts

Isolation of Variables

Isolating the variable is the first step when solving linear equations like \(2 - 3y = 8\). The main idea here is to manipulate the equation in such a way that the variable, in this case, \(y\), stands alone on one side of the equation. Think of it as trying to "get \(y\) by itself" on one side to find its value.To begin, identify which operations are affecting the variable. Here, \(y\) is multiplied by \(-3\) and there is a constant \(2\) on the same side. To isolate \(y\), begin by eliminating these components step-by-step:

• Reverse the effect of the operations acting on the variable by utilizing inverse operations (such as addition to counter subtraction or vice-versa).
• Proceed strategically, first undoing any additions or subtractions, followed by undoing multiplications or divisions last.

Practicing isolation of variables helps build a strong foundation for more complex problem-solving tasks.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to re-arrange and simplify equations. It is a crucial skill that allows you to transform a given equation into a form that is easier to solve.When faced with \(2 - 3y = 8\), algebraic manipulation is necessary to help transition from the original complicated equation to an equation where \(y\) is isolated. Here are the strategic moves made:

• Add \(3y\) to both sides to shift increasingly complicated elements (like negative coefficients) over to the opposite side.
• Ensure any constant terms are on the opposing side away from the variable \(y\) to keep both sides balanced.
• Keep track of signs just as carefully as numbers – errors with positive or negative signs can throw off entire solutions.

Ultimately, careful attention to each small algebraic change leads to a clean and manageable solution.

Equation Simplification

Equation simplification is about reducing an equation to its simplest and most understandable form. Through simplification, we make the final steps towards solving a linear equation more efficient.After isolating the necessary variable \(y\) in \(2 = 8 + 3y\), simplifying starts with streamlining terms:

• Begin by moving constants to one side and variables to the other in the easiest way possible, for example through subtraction or division.
• Every move should aim to reduce the equation bit by bit, from complex to straightforward expressions.
• In this problem, once simplified to \(0 = 2 + y\), simply subtract \(2\) to isolate \(y\).

Simplification aids in not just solving the current problem but building the intuition needed for approaching new and unfamiliar ones, allowing clear pathways to the solution.