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Understanding how to isolate a variable is key to solving linear equations. Isolating the variable means getting the variable by itself on one side of the equation. Let's look at the equation: 4x + 3y = 7.
To isolate y, we need to perform operations that leave y alone on one side. Start by moving all terms that do not include y to the other side. Subtract 4x from both sides:
4x + 3y - 4x = 7 - 4x
This simplifies to: 3y = 7 - 4x.
By isolating the term with y, dealing with the rest of the equation becomes straightforward. Now, the next step is to fully isolate y by dividing both sides by 3.

Linear equations have variables raised only to the first power. They represent straight lines when graphed. In the given exercise, the equation 4x + 3y = 7 is a linear equation. The goal is to simplify the equation to find the variable y.
Linear equations can be written in multiple forms, such as standard form (Ax + By = C) and slope-intercept form (y = mx + b). Recognizing these different forms is important for solving problems and understanding the relationships between variables.
In this problem, we converted the standard form (4x + 3y = 7) to the slope-intercept form by isolating y. This helps to easily interpret how changes in x affect y.

Algebraic manipulation involves moving terms around to simplify an equation. This can include adding, subtracting, multiplying, or dividing both sides of an equation by the same number. Let's review the steps in the problem:
1. Subtract 4x from both sides: 3y = 7 - 4x.
2. Divide both sides by 3 to isolate y: \[ y = \frac{7 - 4x}{3} \]
Each step follows algebraic rules to ensure the equation stays balanced. Practice these steps regularly to become comfortable with manipulating equations. Remember to always perform the same operation to both sides of the equation to keep it balanced.
This skill is essential for solving more complex algebraic problems and forms the foundation for advanced mathematical concepts.