Problem 25 - Elementary Algebra

Question

Solve. $$ -y=13 $$

Step-by-Step Solution

Answer

The solution is \(y = -13\).

Step 1: Identify the equation

The equation given is \[-y = 13\]. This is a simple one-variable equation.

Step 2: Isolate the variable

To solve for \(y\), you need to isolate \(y\) on one side of the equation. Since the variable is \(-y\), you can remove the negative sign by multiplying both sides of the equation by \(-1\). This gives: \[(-1)(-y) = (-1) imes 13\].

Step 3: Solve the equation

After multiplying both sides by \(-1\), the equation simplifies to \[y = -13\]. Therefore, \(y = -13\) is the solution to the equation.

Key Concepts

Understanding One-Variable Equations

A one-variable equation is an equation that involves only one variable, often represented as a letter like \(x\) or \(y\). In these equations, the goal is to find the value of the variable that makes the equation true. These equations can be as simple as \(-y = 13\) or more complex.

With a one-variable equation, there are a few key points to remember:

• Simplicity: Only one variable is involved, making it easier to focus on solving for that single unknown.
• Balance: An equation represents a balance, meaning whatever you do to one side, you must do to the other side to maintain equality.

By clearly understanding these aspects, solving one-variable equations becomes more manageable, as you always strive to find the value of the variable.

Isolation of Variable

Isolating the variable is a crucial step in solving equations. The main idea is to manipulate the equation to have the variable by itself on one side. Let's explore how this works through our example equation \(-y = 13\).

Here’s how you can isolate the variable:

• Focus on the Variable: Identify where the variable is located and what operations are affecting it. In this case, \(-y\) is on the left side.
• Eliminate Coefficients: Remove any numbers or operations around the variable. For \(-y = 13\), you can multiply both sides by \(-1\) to get \(y\) by itself, as it cancels out the negative sign.

By bringing the variable into isolation, you simplify the equation, making it easier to see the solution.

Utilizing the Multiplication Property of Equality

The multiplication property of equality is a powerful tool when solving equations. It states that you can multiply both sides of an equation by the same non-zero number without changing the equation's truth. This property helps in keeping the equation balanced.

In our example \(-y = 13\):

• Apply the Property: Here, multiplying both sides by \(-1\) transforms \(-y\) to \(y\), removing the negative. So, \((-1)(-y) = (-1) \times 13\).
• Solve the Equation: This multiplication simplifies to \(y = -13\).

By using the multiplication property of equality, you've made the equation easier to solve, allowing you to find the value of \(y\). This approach is very handy in equations where you need to eliminate coefficients or change signs.