Problem 8 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(7 y-3=25\)

Step-by-Step Solution

Answer

The solution to the equation \(7y - 3 = 25\) is \(y = 4\).

Step 1: Isolate the y-term on one side

We start by performing an addition operation on both sides of the equation to remove the -3 from the left. This gives us \(7y - 3 + 3 = 25 + 3\), which simplifies to \(7y = 28\).

Step 2: Solve for y

Now, to isolate \(y\), we divide both sides of the equation by \(7\). This gives us \(y = 28 / 7\).

Step 3: Calculate the Result

Finally we calculate the right side of the equation to get the final result: \(y = 4\).

Key Concepts

Understanding Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. These symbols, often represented by letters, stand for numbers whose exact values are not yet known. Algebra is crucial because it forms the foundation of nearly all mathematical principles and is used in everyday life, such as calculating distances, financial planning, and even cooking. In our exercise, we're dealing with a linear equation, which is one of the simplest forms of algebra.

• Linear equations form straight lines when graphed
• They have the general form of \( ax + b = c \)
• In our problem, the equation is \(7y - 3 = 25\)

Mastering algebra requires a strong understanding of how to perform operations to simplify and solve equations. This involves:

• Identifying the terms and coefficients
• Using arithmetic operations to isolate variables
• Applying inverse operations to solve for unknowns

By understanding these key concepts, you'll be well-equipped to tackle a wide variety of algebraic problems.

Solving One-Variable Equations

One-variable equations, also known as linear equations in one variable, involve solving for an unknown that appears only once. These equations have the standard form \(ax + b = c\), where we solve for \(x\), or in our case, \(y\). Comprehending how to manipulate these equations to find the unknown variable is essential.

• The goal is to isolate the variable
• Use addition or subtraction to eliminate constants
• Use multiplication or division to adjust coefficients

In the given example, \(7y - 3 = 25\), the process involves reversing each operation applied to \(y\) until it stands alone:

• Add 3 to both sides to remove the negative 3
• Divide by 7 to solve for \(y\)
• Result: \(y = 4\)

Understanding these basic operations will allow you to solve any similar linear equation effectively. Remember, the operations you perform should maintain the balance of the equation.

The Power of Step-by-Step Solutions

Step-by-step solutions help break down the process of solving equations into manageable parts. This structured approach makes challenging problems seem more approachable and definitely helps in understanding the logical flow of the solution.

• Clearly identify each move or operation
• Understand why each step is justified
• Ensure the final answer makes sense in context

For our problem, the step-by-step process comprises:1. Remove the constant term (-3) by adding it to both sides: Transform the equation to \(7y = 28\).2. Solve for \(y\) by dividing both sides by the coefficient (7): Results in \(y = 4\).3. Verify the final solution to ensure that the initial equation holds true.By following each step diligently, you can ensure no mistakes are made. This method instills confidence and clarity, reinforcing learning and understanding. For students learning algebra, this step-by-step method is invaluable.