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Algebraic equations are mathematical statements that assert the equality of two expressions. In essence, they are like a balance scale; whatever is done to one side must also be done to the other to maintain equality. An equation is composed of variables, coefficients, constants, and arithmetic operators. In the equation \( -17x = -51 \) from the exercise, \( x \) is the variable we need to find, -17 is the coefficient of \( x \) indicating how many times \( x \) is multiplied, and -51 is a constant.

Understanding algebraic equations is crucial as they form the basis for a wide range of mathematical concepts and problem solving in science, engineering, economics, and beyond. They allow us to represent real-world problems in a way that we can manipulate mathematically to find a solution.

To isolate the variable means to rearrange the equation so that the variable stands alone on one side of the equation, facilitating the solution for its value. This involves performing operations that 'undo' the effect of the coefficients and constants on the variable.

In the given problem, isolation is achieved by dividing both sides of the equation by -17. This is because -17 is multiplying the variable \( x \), and dividing by -17 is the opposite or 'inverse' operation, effectively cancelling it out. The goal is to get \( x = \) something, which indicates that the variable has been successfully isolated.

Steps to Isolate the Variable:

• Determine what is being done to the variable.
• Perform the inverse operation to 'undo' what's being done.
• Apply this operation equally to both sides of the equation.
• Continue until the variable stands alone.

This process is a core part of solving algebraic equations and is an essential skill in algebra.

Solving an equation systematically can be broken down into a series of steps that, when followed properly, lead to the solution. The given exercise demonstrates this process well, with a simple linear equation.

Key Steps for Solving Equations:

• Set up the equation and understand its form.
• Identify the variable to be solved for and begin the process to isolate the variable.
• Perform the necessary mathematical operations on both sides of the equation.
• Obtain the value of the variable and simplify if necessary.
• Check the solution by substituting it back into the original equation.

By following these steps, you can solve most basic algebraic equations accurately. Checking the solution is important as it confirms the validity of the result and reinforces the concept that what is performed on one side of the equation must also be applied to the other to maintain balance.