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Algebraic manipulation is the backbone of solving equations, as it allows us to rearrange and simplify expressions to isolate variables. In the given problem, our first step of algebraic manipulation involves moving terms containing the variable \(x\) onto one side of the equation while leaving constants on the other side. This operation rewrites the equation, making it easier to identify and solve for \(x\).When performing algebraic manipulation, it's essential to maintain the equation's balance. Whatever you do on one side, you must do to the other. This rule ensures that the equation remains correct and equal. Keep in mind:

• When you add or subtract terms, do it equally on both sides.
• If you need to multiply or divide, apply the operation throughout the entire equation.

By understanding and following these rules, you can confidently solve linear equations that may initially appear complex.

Factoring is an essential algebraic skill that simplifies expressions, making it possible to find solutions or identify common components. In this exercise, we need to factor out \(x\) from the terms on the left side of the equation \(7bx + 2cx\). The purpose of factoring here is to express the left-hand side as a product of \(x\) and another factor.To factor \(x\) out:

• Identify the common variable or number across terms, which is \(x\) in our case.
• Rewrite the expression as a multiplication of this common factor and the remaining parts.

In practice, after factoring \(x\) from \(7bx + 2cx\), we have \(x(7b + 2c)\). This step simplifies the equation considerably, giving us a clear path to solving for \(x\) by addressing a straightforward equation instead of managing multiple terms at once.

The final stage in solving any linear equation is isolating the target variable, in this case \(x\), to find its value. Once you've performed algebraic manipulation and factoring, your equation becomes simpler, like having \(x(7b + 2c) = 4a\).To solve for \(x\):

• Recognize that \(x\) is being multiplied by some factor, \(7b + 2c\).
• Reverse this multiplication by dividing both sides of the equation by the factor, \(7b + 2c\).

Thus, the equation becomes \(x = \frac{4a}{7b + 2c}\), and we have successfully isolated \(x\).This approach highlights that any multiplication can be reversed through division, a critical aspect of equation solving. Consistently using this knowledge ensures that you can solve various linear equations, regardless of their initial presentation.