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The addition method, also known as the elimination method, is a technique used to solve systems of linear equations. The goal is to eliminate one of the variables by adding or subtracting the equations. This allows us to solve for the remaining variable. Let's break it down:

First, align the equations in a compatible format. Ensure both equations are simplified and terms are lined up. Multiply each equation by a constant if necessary, to make the coefficients of either variable the same (but opposite for addition). This allows for one of the variables to be eliminated when the equations are added together.

For our example, we work to eliminate fractions early on (Step 1), then bring the coefficients of one of the variables to the point where they can be added up to eliminate that variable (Step 4). The simplified equations are then added (Step 5), providing a single variable equation that can be easily solved. Finally, the value obtained is substituted back into one of the original equations to get the value of the other variable (Step 7). This method is particularly useful when the system is not straightforward to solve by substitution.

An independent system of equations has exactly one unique solution. This means that the lines represented by the equations intersect at a single point. If a system's equations are consistent and have different slopes, they will meet at one point.

In our problem, after using the addition method to eliminate fractions and solve the variables, we found the unique solution \( x = 4 \) and \( y = 6 \). Therefore, the system is classified as independent.

Independent systems are characterized by:

• Unique solutions
• Non-parallel lines
• Different slopes in linear equations

Recognizing an independent system can help you understand that the pair of equations won't endlessly circle around without intersecting or overlap completely.

Solving equations with fractions can be tricky, but it becomes simpler once you eliminate fractions. This involves transforming all terms to integer values.

To eliminate fractions:

• Find the least common multiple (LCM) of the denominators in the equations.
• Multiply every term in the equation by this LCM to clear the fractions.

In our exercise, the first equation \( \frac{x}{2}+\frac{y}{2} = 5 \) was multiplied by 2 to give \( x + y = 10 \). Similarly, multiplying the second equation \( \frac{3x}{2}-\frac{2y}{3}=2 \) by 6 eliminated the fractions, resulting in \( 9x - 4y = 12 \).

This process not only makes the equations easier to handle but also paves the way for subsequent steps in the addition method, ultimately leading to the solution.