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In algebra, solving equations is a fundamental skill. The goal is to find the value of the variable that makes the equation true. You start with a given equation and perform operations on both sides of the equation to isolate the variable. This means you’ll often be adding, subtracting, multiplying, or dividing terms on both sides to keep the equation balanced. When solving linear equations, you are dealing with equations where the variables are raised to the first power (like x or y) and no variable is multiplied by another variable. Let's dig deeper into how we solve such equations step-by-step. Always remember that any operation you perform on one side must also be performed on the other side.
Below are some common operations you might perform:

• Adding or subtracting the same quantity from both sides
• Multiplying or dividing both sides by the same quantity (except for zero)

Isolating the variable is a critical step in solving equations. This means getting the variable by itself on one side of the equation. In the example, we needed to solve for y in the equation 3x + 2y = 11. We start by isolating the term with y. To do this, we subtract 3x from both sides of the equation to get 2y on one side. This gives us 2y = 11 - 3x.
This step is important because it simplifies the equation and brings us one step closer to finding the value of y.
Follow these steps to isolate a variable:

• Identify the term with the variable you need to isolate.
• Use addition or subtraction to move other terms to the opposite side of the equation.

Once you have isolated the term with the variable, the next step is to solve for it.

Algebraic manipulation involves using algebraic rules and operations to simplify or solve equations. In our example, after isolating the term 2y, we perform algebraic manipulation by dividing both sides by 2. This action allows us to solve for y. So, from 2y = 11 - 3x, we divide everything by 2 to get y = (11 - 3x) / 2.
This is a key use of algebraic manipulation:

• Identify the operation involving your variable.
• Perform the opposite operation to isolate the variable.
• Simplify the equation step-by-step to keep it manageable.

By understanding and applying these manipulation techniques, solving even complex equations becomes more straightforward and systematic.