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The process of isolating a variable, like many things in life, works best when you take it step by step. Isolating a variable means to get the variable alone on one side of an equation, with everything else on the other side. This technique is vital for solving equations because it allows you to find out what value the variable must take on. Take the example from the exercise, such as in equation a: - We started with \( y + 2 = 5x \). - To isolate \( y \), we need to get rid of the \( +2 \) that is currently keeping \( y \) from being alone. - By subtracting 2 from both sides of the equation, we maintain the balance of the equation but only \( y \) is left on the left side. The key here is consistency. Whatever operation you do to one side of the equation; do the same to the other side. This keeps the equation true and valid as you manipulate it. After you have successfully isolated the variable \( y \), you have succeeded in solving the part of the equation you were working on.

Algebraic manipulation involves using mathematical operations to rearrange and simplify equations or expressions. To handle algebraic manipulation, follow these steps:

• Identify the operations affecting the variable.
• Reverse those operations using inverse functions (like addition and subtraction, or multiplication and division).
• Perform the same operations on both sides of the equation to maintain balance.

In equation b: - Start with \( 5y = 4 - 7x \). - Here, \( y \) is affected by multiplication (5 times \( y \)). To reverse this, divide the whole equation by 5. - This operation simplifies the equation to \( y = \frac{4 - 7x}{5} \).By applying algebraic manipulation, you simplify complex equations and make them easier to understand or solve. It's like untangling a knot, step by step.

Equation balancing is an essential concept needed to solve equations correctly, ensuring each side of the equation remains equal as you manipulate it. This concept hinges on the idea of equal treatment.Every action you perform on one side of an equation must be mirrored on the other side. Here's how you can maintain balance in equations:

• Add or subtract the same amount on both sides.
• Multiply or divide the entire equation by the same quantity.

For equation c:- Start with \( 2y - 4x = 3 \).- We add \( 4x \) to both sides to ensure the equation remains balanced, giving us \( 2y = 4x + 3 \).- Follow up by dividing everything by 2 to finally solve for \( y \), resulting in \( y = 2x + \frac{3}{2} \).Balancing keeps the integrity of equations intact, making sure that throughout your manipulations, the equality holds true. Think of it as a balancing act in a performance – every change must be countered by another change to keep everything level.