91Ó°ÊÓ

Solving equations means finding the value of the variable that makes the equation true. A linear equation involves variables that are only to the first power, which means they have no exponents or radicals applied to them. The aim is to simplify these equations until you can clearly see the solution.
To solve an equation, you follow a logical sequence of operations to isolate the variable. These operations must keep the equation balanced, meaning whatever you do to one side of the equation, you must do to the other. This is to preserve equality.
To solve the equation \(8.6y + 3.4 = 4.2y - 9.8\), we worked to both manipulate and balance the equation, ultimately finding the value of \(y\) where both sides of the equation are equal.

Algebraic manipulation involves changing the structure of an equation using various operations such as addition, subtraction, multiplication, and division. These actions help simplify the equation, so the solution becomes clear.
In our original problem, we utilize algebraic manipulation to make terms more manageable:

• First, we subtract \(4.2y\) from both sides. This brings terms involving \(y\) together, simplifying to \(4.4y + 3.4 = -9.8\).


• Next, subtract \(3.4\) from both sides. This isolates terms involving \(y\), simplifying further to \(4.4y = -13.2\).

These operations allow a complicated equation to become more straightforward, paving the way to discover the solution through simpler calculations.

Isolation of a variable involves keeping the variable by itself on one side of the equation. This makes it easier to see what value the variable must have to satisfy the equation. It's a crucial technique in solving linear equations.
Once algebraic manipulations have brought terms involving \(y\) together, our goal becomes clear: make \(y\) the subject of the equation.
We achieve this by dividing both sides of the simplified equation \(4.4y = -13.2\) by \(4.4\). This breaks down to:

• Divide \(-13.2\) by \(4.4\), resulting in \(y = -3\).

Once the variable is isolated, you can directly see the solution, knowing that your steps maintained balance and correctness throughout the process. This tactic of isolating the variable is a powerful algebraic tool that simplifies solving linear equations.