Problem 99 - Intermediate Algebra

Question

Solve each equation. $$ 13.5 y+16.2=0 $$

Step-by-Step Solution

Answer

The solution to the equation is \(y \approx -1.2\).

Step 1: Isolate the Variable Term

First, we need to isolate the term with the variable \(y\). To do this, we subtract 16.2 from both sides of the equation:\[13.5y + 16.2 - 16.2 = 0 - 16.2\] This simplifies to:\[13.5y = -16.2\]

Step 2: Solve for the Variable

Now, divide both sides of the equation by 13.5 to solve for \(y\):\[y = \frac{-16.2}{13.5}\] Calculating the right-hand side gives:\[y \approx -1.2\]

Key Concepts

Isolate the Variable

In solving linear equations, one of the crucial steps is to isolate the variable. This means we want the unknown variable, like our "y" in the equation \(13.5y + 16.2 = 0\), to be by itself on one side of the equation. When the variable stands alone, it's easier to find out its value.

To isolate the variable, we perform operations that effectively 'move' other numbers away from it. Think of it like unpacking a suitcase—removing layers until you reach the core item you care about. In equations, these "layers" are any extra terms that aren't attached to the variable by multiplication or division.

• Identify the term with the variable. In our example, that's \(13.5y\).
• Perform the opposite operation to remove constants from the side of the equation that contains the variable.

In this case, we subtracted \(16.2\) from both sides, simplifying the equation to \(13.5y = -16.2\).

Arithmetic Operations

Arithmetic operations are the basic building blocks of math that include addition, subtraction, multiplication, and division. To solve linear equations, we often need these operations to manipulate and simplify expressions in equations. Understanding how to apply these correctly helps in shifting terms and isolating variables.

For solving equations, we often focus on:

• Addition and Subtraction: Used for moving constants or terms from one side of the equation to the other. If you need to remove a positive number from one side, you subtract it. If the number is negative, add it instead.
• Multiplication and Division: Useful for adjusting the coefficients attached to variables. If a variable is multiplied by a number, you can divide by that number to simplify.

In the original exercise, we used subtraction to remove \(16.2\), making the equation easier by focusing solely on the \(13.5y\) term, which is essential for the next steps. Understanding how these operations interact is key to manipulating equations successfully.

Algebraic Manipulation

Algebraic manipulation refers to rearranging and simplifying equations to make them easier to solve. Once the variable is isolated, algebraic manipulation helps in deriving the final answer. It's all about applying basic arithmetic rules to mold the equation into a simple "variable equals value" format.

In context:

• After isolating \(y\), you simplify the equation by performing operations that clear coefficients attached to the variable.
• For instance, \(13.5y = -16.2\) involves dividing both sides by \(13.5\) to find \(y\).

This division step gives more clarity, yielding \(y \approx -1.2\). Algebraic manipulation turns complex expressions into manageable solutions, providing a straightforward path to the answer. Understanding how to rearrange and solve equations is a fundamental skill in algebra that builds a strong foundation for more advanced mathematics.