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In algebra, the substitution method is a fundamental technique used to solve equations or systems of equations. This approach is highly valuable when you have an equation with a variable and you want to find out if a specific value is a solution.

The process involves taking the given number and replacing the variable with it. As demonstrated in the exercise, the number 3 is substituted for the variable 'k' in the equation, resulting in the new equation: \(3(3) + 5 = 5(3) - 1\).

This technique not only allows us to find a potential solution but also to simplify complex equations into a more manageable form. It's essential to understand the substitution process meticulously because, at times, encountering variables in numerous places in an equation can lead to errors if the substitution is not carried out consistently. Picturing the variable as a placeholder that holds the space for a number helps in executing this method correctly.

Algebraic manipulation refers to the use of various algebraic operations to rearrange and simplify equations. This skill is critical in solving algebra problems effectively. The manipulation includes operations such as adding, subtracting, multiplying, and dividing both sides of the equation by numbers and variables, as well as applying the distributive property.

In the context of the provided example, once we substitute '3' for 'k', the next step involves simplification by multiplying out the brackets and combining like terms: \(9 + 5 = 14\) and \(15 - 1 = 14\).

Consistent practice in algebraic manipulation bolsters a student's ability to handle more intricate equations. It's not just about applying the operations but also knowing when and in what order to use them to make the problem more accessible. Proper manipulation can often reveal solutions that are not immediately evident from the original form of the equation.

Equation simplification is the process used to condense an equation into its simplest form, making it easier to interpret and solve. The goal here is to clear away the clutter of the equation to reveal the core relationship between the variables.

In the given exercise, after the substitution step, simplification involves working out the arithmetic inside the brackets and combining similar terms to find that both sides equal 14.

Why Simplify?

Simplifying an equation can sometimes be the difference between recognizing a solution and being perplexed by an algebraic mess. Seeing the equation reduce to such simple terms as \(14 = 14\) offers a clear verification that the substitution is indeed correct.

Simplification is not just about making life easier; it represents a foundational skill in mathematics that aids in better understanding and solving more complex problems in the future.