Problem 7 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the given equation. $$ -2 y+3=-7 $$

Step-by-Step Solution

Answer

The solution to the given linear equation \(-2y + 3 = -7\) is \(y = 5\).

Step 1: Subtract 3 from both sides of the equation

Subtract 3 from both sides in order to get rid of the constant on the left side of the equation. This will give us the following new equation: \[ -2y = -7 - 3 \]

Step 2: Simplify the right side of the equation

Perform the subtraction on the right side of the equation: \[ -2y = -10 \]

Step 3: Divide both sides by -2

In order to isolate y, divide both sides of the equation by -2: \[ y = \frac{-10}{-2} \]

Step 4: Simplify and find the value of y

Simplify the fraction by dividing -10 by -2: \[ y = 5 \] As a result, y equals 5, and the solution to the given linear equation is y = 5.

Key Concepts

Solving Linear Equations

Solving linear equations like \(-2y + 3 = -7\) is all about finding the value of the variable that makes the equation true. In this equation, that variable is \(y\). Our goal is to perform operations that transform the equation step by step until we isolate \(y\) on one side. This involves moving terms around with algebraic manipulation and then simplifying parts of the equation until it's solved. Let's explore this process, starting with understanding the key components:

• Linear Equation: An equation that can be written in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
• Variable: In our case, \(y\) is the unknown we are trying to solve for.
• Operations: Adding, subtracting, multiplying, or dividing both sides of the equation by the same number to keep the equation balanced.

Remember, the theme here is balance—every change you make to one side, you must do to the other, keeping the equation's integrity intact. Let's delve deeper into the tools we use for solving equations with algebraic manipulation.

Algebraic Manipulation

Algebraic manipulation refers to the series of arithmetic operations and rearrangements we perform in equations to simplify and solve them. Imagine you're a detective, and your job is to uncover the value of the variable by untangling the equation's parts. In our example:

We first subtracted 3 from both sides of \(-2y + 3 = -7\) to eliminate the constant term attached to the variable. This step simplifies the equation to \-2y = -10\.

• Elimination of constants: Subtracting 3 removed the constant from the left side.
• Arithmetic Operations: We then subtracted 10 (the result of -7 - 3) to achieve a simpler equation.

These transformations are algebraic manipulations because they systematically simplify the equation while maintaining equality. It's like peeling layers of an onion— you gradually simplify until you can see the core, which is your variable.

Next, let's move on to isolating the variable to discover its value.

Isolation of Variables

Isolation of variables is the final stage of solving an equation, where our goal is to have the variable completely on one side, standing alone with its value. This isolation allows us to see directly what the variable equals. In the equation \-2y = -10\, our task is to isolate \(y\) by eliminating \-2\ before the \(y\).

• Division: By dividing both sides by \-2\, we effectively cancel out the multiplier attached to \(y\).

\[y = \frac{-10}{-2} = 5\]We perform the division to simplify our expression and thus isolate \(y\). At this point, we get \(y = 5\). The process of isolation not only solves the problem but also shows the power of balance in equations. Every manipulation focuses on maintaining equality while uncovering the variable's identity. Now, having \(y\) equal to 5 reveals the equation's solution.