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Equation solving is a fundamental part of algebra and mathematics in general. It involves finding the value of the variable that makes the equation true. In our example, the equation given is:\[ 7(2x + 5y) = 6 \]This equation contains both "x" and "y" variables. Our task here is to solve for "y". Solving equations usually involves several steps. Sometimes, these steps include simplifying the equation, using different algebraic properties, or rearranging terms to isolate the variable we are solving for. We ensure that the equality is maintained across both sides of the equation throughout the process.

The distributive property is an essential algebraic property used to simplify expressions and equations. This property states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. For example:\[ a(b + c) = ab + ac \]In our equation, \( 7(2x + 5y) = 6 \), we apply the distributive property. This means we multiply 7 by each term inside the parenthesis:- Multiply 7 by \(2x\) to get \(14x\)- Multiply 7 by \(5y\) to get \(35y\)After applying the distributive property, our equation looks like this: \[ 14x + 35y = 6 \]This step is crucial as it helps simplify complex expressions, allowing us to solve equations in a more straightforward manner.

To isolate a variable means to get it by itself on one side of the equation. This is a key step in solving equations when you need to find the value of a specific variable. In our example, we need to isolate "y" from the equation:\[ 14x + 35y = 6 \]First, we subtract \(14x\) from both sides to move all terms containing "x" to the opposite side:\[ 35y = 6 - 14x \]Next, to completely isolate \(y\), we divide every term by 35:\[ y = \frac{6 - 14x}{35} \]This step ensures that "y" is on its own and expressed in terms of "x". Isolating a variable helps to identify its relationship to other variables and constants within the equation. Understanding and practicing this concept is essential for mastering algebra and solving more complex equations.