Problem 10 - Elementary Algebra

Question

Solve. $$ -2 y-52=3 y+13 $$

Step-by-Step Solution

Answer

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

Step 1: Gather Like Terms

Start by getting all terms involving \( y \) on one side of the equation. Subtract \( 3y \) from both sides to isolate the terms with \( y \): \(-2y - 3y - 52 = 13\). This simplifies to \(-5y - 52 = 13\).

Step 2: Isolate the Variable Term

Add 52 to both sides to isolate the term with \( y \): \(-5y - 52 + 52 = 13 + 52\). This simplifies to \(-5y = 65\).

Step 3: Solve for \( y \)

Divide both sides by \(-5\) to solve for \( y \): \( y = \frac{65}{-5} \). Simplifying this gives \( y = -13 \).

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of algebra, and it means finding the value of the unknown variable that makes the equation true. In the context of the given exercise, the goal is to determine the value of \( y \) that satisfies the equation \(-2y-52=3y+13\). To solve an equation, you often need to perform several algebraic steps, such as gathering like terms and isolating variables. Breaking down each step into more manageable pieces can make the problem more straightforward. Remember, the objective is always to simplify the equation until \( y \) is on its own, revealing its true value.

Like Terms

Like terms are terms that have the same variable raised to the same power. In algebra, combining like terms simplifies expressions and equations. Consider \(-2y\) and \(3y\) in the exercise. Both have the variable \( y \), thus they are like terms. To simplify, it's typical to gather all the like terms on one side of the equation.
This helps in reducing the complexity, leading eventually to an easier solution. Here's an easy way to recognize them: they must look alike in terms of variables and their powers, though coefficients can differ.

• Example: \(-2y\) and \(3y\) are like terms because they share the variable \( y \).
• Non-example: \(2y^2\) and \(2x\) are not like terms because of differing variables and powers.

Proper identification and grouping of like terms are vital in simplifying and solving algebraic equations efficiently.

Isolating Variables

To solve an equation efficiently, isolating the variable means getting the variable by itself on one side of the equation. This concept is pivotal. Once you've combined like terms, focus shifts to getting the variable term, like \(-5y\) in our exercise, completely isolated.
Isolation forms the equation into \(-5y=65\), making it straightforward to determine \( y \) by performing a simple arithmetic operation. Often, isolation requires operations like addition, subtraction, multiplication, or division. The process is like peeling layers off an onion to reach the core of the solution.
One helpful tip: Always perform the same operation on both sides of the equation to maintain its balance.

Simplifying Expressions

Simplifying expressions in equations means making the equation or expression easier to work with by reducing it to its simplest form. In the given problem, after isolating the \( y \), you simplify \(-5y = 65\) by dividing both sides by \(-5\). This step shows the process of drilling down through complexities to reveal the answer.

• The division operation in this context transforms \(-5y = 65\) into \(y = \frac{65}{-5}\).
• Finally, simplifying \(\frac{65}{-5}\) produces the result \(y = -13\).

Simplification makes the solutions more understandable and clear, ready for real-world application in varied algebra problems. It's essentially the art of making mathematics more approachable by removing complications.