91Ó°ÊÓ

Linear equations are fundamental in algebra and serve as a springboard for understanding more complex concepts. A linear equation is any equation that can be rearranged in the form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. Solving linear equations involves finding the value of the variable that makes the equation true.
To solve a linear equation, you need to isolate the variable, usually \(x\), on one side of the equation while maintaining balance. This means whatever operations you perform on one side of the equation, you must also perform on the other. This concept is crucial as it ensures the equation's equality is maintained.
A common method to solve linear equations is to use basic arithmetic operations to simplify the equation step by step until you find the solution.

Isolating the variable is a critical step in solving any equation. The primary aim is to "capture" the variable on one side of the equation while all the numbers or constants are on the opposite side.
In the exercise involving the equation \(7x - 5 = 72\), the first step is to neutralize the \(-5\) by adding 5 to both sides. This gives:

• \(7x = 72 + 5\)
• \(7x = 77\)

By adding 5 to both sides, we effectively isolate the term containing \(x\).
Now, \(7x\) is standing alone on one side, making it easier to tackle the final step of completely freeing \(x\) from its coefficient.

Algebra can often feel like a puzzle where you use known pieces, like numbers and operations, to uncover the unknown, often represented by variables. It is a branch of mathematics that uses symbols to represent numbers and relationships between them.
In algebra, solving equations is one of the foundational skills that allow you to manipulate and solve expressions to find unknown values. It requires understanding operations like addition, subtraction, multiplication, and division and how these can alter an equation.
Every step in solving an equation involves an operation that simplifies the problem. For example, once we isolated \(7x\) in the given problem, we solved \(x\) by dividing both sides by 7:

• \(x = 77 / 7\)
• \(x = 11\)

This process demonstrates how algebra allows us to creatively yet logically simplify and solve mathematical problems by breaking them down into more manageable steps.