Problem 37 - Elementary Algebra

Question

Solve. $$ y+23=25 $$

Step-by-Step Solution

Answer

\( y = 2 \)

Step 1: Identify the Equation

We are given the equation \( y + 23 = 25 \). The goal is to find the value of \( y \) that makes this equation true.

Step 2: Isolate the Variable

To solve for \( y \), we need to isolate it on one side of the equation. We can do this by subtracting 23 from both sides of the equation. This gives us: \( y + 23 - 23 = 25 - 23 \).

Step 3: Simplify Both Sides

Simplify both sides of the equation after subtraction: \( y = 2 \). This means that \( y \) equals 2.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra, and it serves as the basis for many mathematical concepts. An equation is simply a mathematical statement that asserts the equality of two expressions. In our example, we have the equation \( y + 23 = 25 \). The primary goal is to find the value of the variable, in this case \( y \), that makes the equation true.

One important thing to remember is that whatever operation you perform on one side of the equation, you must perform the same operation on the other side. This balance is crucial in maintaining the equality. By doing this consistently, you work towards finding a solution that satisfies the equation.

Solving equations also involves identifying the type of equation we are dealing with: linear, quadratic, etc. In this case, it is a linear equation because it only involves powers of the variable \( y \) up to the first degree. Understanding the type of equation can help determine the best method to solve it effectively.

Isolating Variables

The process of isolating a variable is a key step when solving equations, particularly linear ones. The idea is to get your variable (in this case \( y \)) alone on one side of the equation, with all constants or other variables on the opposite side.

Here’s how we do it for the equation \( y + 23 = 25 \):

• Identify the operations affecting the variable. For \( y + 23 = 25 \), it's clear that 23 is added to \( y \).
• Perform the inverse operation to eliminate the constant (or other terms) affecting the variable. Since 23 is added, we'll subtract 23 from both sides to isolate \( y \).

This results in \( y + 23 - 23 = 25 - 23 \), which simplifies to \( y = 2 \).

Isolating the variable is like clearing space around \( y \). Once isolated, it becomes easier to see what \( y \) equals, leading directly to the solution.

Simplifying Equations

Simplifying equations is the process of making an equation easier to solve or understand. It often involves combining like terms, reducing fractions, or clearing out unnecessary elements. In our specific example \( y + 23 - 23 = 25 - 23 \), the equation simplifies to \( y = 2 \).

The primary simplification here was

• Subtracting 23 from both sides to remove it from the left side of the equation.
• This left \( y \) alone, simplifying the equation to the form \( y = 2 \).

Simplification helps in making calculations easy to handle and checking the correctness straightforward. The less complex the equation, the fewer operations there are to potentially introduce errors. In many cases, the simpler a problem, the easier it is to solve quickly and accurately.