Problem 16 - Elementary Algebra

Question

Solve the equations. $$ -y=-10 $$

Step-by-Step Solution

Answer

Answer: The value of y in the given equation is 10.

Step 1: Understanding the given equation

We are given the equation: $$ -y = -10 $$ Our goal is to solve for the variable, y.

Step 2: Isolate the variable y

To isolate y, we need to get rid of the negative sign in front of it. We can do this by multiplying both sides of the equation by -1: $$ (-1) \cdot (-y) = (-1) \cdot (-10) $$ This simplifies to: $$ y = 10 $$

Step 3: Write the final answer

We have successfully isolated the variable y: $$ y=10 $$ So, the solution to the given equation is y = 10.

Key Concepts

Isolating Variables

When solving equations, one of the primary goals is to isolate the variable you are solving for. This means getting the variable by itself on one side of the equation. In the equation \(-y = -10\), the variable we want to isolate is \(y\).
To do this effectively, you should:

• Identify the variable you need to isolate. Here, it's \(y\).
• Move any terms or coefficients away from the variable. This involves algebraic manipulation to reverse operations applied to the variable.
• Remember that whatever operation you do to one side of the equation, you should also do to the other side to maintain equality.

By isolating the variable, you can transform the equation into a simpler form where the solution is evident.

Negative Coefficients

Dealing with negative coefficients can sometimes be confusing for students, but it’s an important aspect of solving linear equations. A coefficient is a number that multiplies the variable, and in \(-y = -10\), \(-1\) is the coefficient of \(y\).
Here's how to handle them:

• To remove a negative sign in front of a coefficient, you can multiply both sides of the equation by \(-1\). This reverses the sign of both sides, allowing you to see the positive counterpart of the equation.
• For example, starting with \(-y = -10\), multiplying both sides by \(-1\) gives you \(y = 10\), effectively neutralizing the negative sign.

Handling negative coefficients correctly is crucial for simplifying the equation and finding the accurate solution.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to rearrange and simplify equations, making it easier to solve for unknown variables.
In our example, \(-y = -10\), some of the essential manipulations are:

• Identify operations applied to the variable and determine the opposite operation. For \(-y\), multiplying by \(-1\) gives \(y\).
• Apply the opposite operation uniformly across the equation, ensuring both sides are treated equally. This keeps the equation balanced.
• Simplify the equation by performing calculations, such as combining like terms if applicable, to reach the solution.

Through practiced algebraic manipulation, students can tackle even the most complex equations with confidence, arriving at the correct answer systematically.