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Simultaneous equations, also known as systems of equations, involve finding values for variables that satisfy multiple equations at once. These equations are solved together rather than individually. In this problem, we are asked to solve the following system of equations:

• \(y = 6x + 2\)
• \(12x - 2y = -4\)

When working with simultaneous equations, the key is to find a set of values for the variables involved—often \(x\) and \(y\)—that make all the equations true simultaneously. It's like solving a puzzle where the pieces must fit together perfectly in any possible combination.

Typically, solving such a system involves finding one variable in terms of the other or using techniques like elimination or substitution to find the precise values. In this case, these methods aim to simplify the equations to a point where looking at the solution becomes evident.

The substitution method is one of the most effective ways to solve systems of equations. It involves expressing one variable in terms of another from one equation and then substituting this expression into another equation. This helps in reducing a system with multiple variables into one with a single variable.

In this exercise, from the equation \(y = 6x + 2\), \(y\) is already expressed in terms of \(x\). We substitute this expression into the second equation:

• \(12x - 2(6x + 2) = -4\)

Simplifying this gives us an equation that we can solve easily. This reduction is crucial because it allows us to focus solely on solving for \(x\) first, rather than dealing with both variables at the same time.

The substitution method is particularly beneficial when one of the equations is already solved or easily solved for one of the variables. Once you have the solution for the substituted variable, you can often quickly find the value of the other variable by going back to any equation from the original set.

Infinite solutions in a system of equations indicate that the equations are dependent, meaning they graphically represent the same line. Each equation has all its solutions in common with the other.

After substituting \(y = 6x + 2\) into the second equation and simplifying, we obtained:

• \(-4 = -4\)

This tells us our solutions don't depend on a specific value of \(x\); instead, they hold true without any restriction on \(x\), indicating infinite solutions.

Infinite solutions occur when the equations in a system are equivalent or multiples of each other. It suggests there isn't just one solution pair that satisfies both equations, but rather a whole line of them. This happens because both equations describe the same line on a graph, so they intersect at infinitely many points along that line.

Algebraic equations involve variables and constants connected through a variety of operations such as addition, multiplication, and subtraction. Solving algebraic equations requires manipulating these numbers to find the values of unknowns.

In this system:

• \(y = 6x + 2\)
• \(12x - 2y = -4\)

Both equations represent algebraic relationships where we need to employ strategies to find \(x\) and \(y\).

Using algebraic manipulation, such as distributing, combining like terms, and isolating variables, allows us to break down complex equations into simpler forms. This simplification is key in solving the system effectively. Proper understanding of how to handle algebraic equations is critical for successfully tackling mathematical problems involving any unknowns.

Algebra forms the basis for much of mathematics, making it essential to learn and master these techniques to solve equations in everyday mathematical tasks and more specialized fields like engineering and sciences.