Problem 101 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ y+5=-7 $$

Step-by-Step Solution

Answer

The solution is \( y = -12 \).

Step 1: Identify the Equation

The given equation is \( y + 5 = -7 \). Our goal is to find the value of \( y \) that satisfies this equation.

Step 2: Isolate the Variable

To solve for \( y \), we need to get \( y \) by itself on one side of the equation. We can do this by subtracting 5 from both sides.

Step 3: Perform the Subtraction

Subtract 5 from both sides of the equation: \( y + 5 - 5 = -7 - 5 \). This simplifies to \( y = -12 \).

Key Concepts

Isolation of Variables

One of the fundamental methods for solving algebraic equations is isolating the variable. This technique involves manipulating the equation in such a way that the variable of interest is "isolated" on one side. This allows us to find its value more easily.
In our example, the equation is given as \( y + 5 = -7 \). Our goal is to determine what value of \( y \) would make this equation true. In order to isolate \( y \), we need to remove anything that is added or subtracted to it.
The process usually involves reversing operations that are performed on the variable. If something is added to the variable, we subtract it from both sides of the equation to isolate the variable. Similarly, if something is subtracted, we add it to both sides. This is crucial in moving towards the solution.

Subtraction Property of Equality

The subtraction property of equality states that if you subtract the same value from both sides of an equation, the equality is still maintained. This property stems from the fact that an equation is like a balance scale. Whatever you do to one side must be done to the other to keep it balanced.
In practice, this means that in the equation \( y + 5 = -7 \), we can subtract 5 from both sides to help isolate \( y \).
- Start with: \( y + 5 = -7 \)- Apply the subtraction property: \( y + 5 - 5 = -7 - 5 \)- Simplify: \( y = -12 \).
By performing this operation, we have not changed the equality, only transformed the equation into a simpler form where the solution is evident.

Basic Algebra Concepts

Understanding the basics of algebra is key for solving equations. Algebra often involves finding unknown values (variables) that satisfy a given condition (equation). Here are some essential points:

• Variables represent unknown numbers and are often denoted by letters like \( y \), \( x \), etc.
• Operations like addition, subtraction, multiplication, and division are used to describe relationships between variables and numbers.
• Solving an equation means finding all the values that make the equation true.

In our specific example, \( y + 5 = -7 \), we employ basic algebra to find that \( y = -12 \). Solving involves logical steps that ensure we maintain balance in the equation while unriddling the value of \( y \).
Remember, patience and practice are crucial in mastering these concepts, as they form the foundation for more advanced mathematical topics.