Understanding the Problem: Finding the Value of x in a System of Equations

The question asks us to find the value of the variable 'x' in a given system of two linear equations. A system of equations is a set of two or more equations that contain the same variables. The solution to a system is the set of values for the variables that satisfy all equations in the system simultaneously.

The System of Equations Given

We are provided with the following two equations:

1. \(\frac{2x}{3} + \frac{y}{2} = 4\)
2. \(\frac{x}{3} - \frac{y}{2} = 1\)

Solving the System of Equations: Using the Elimination Method

To find the value of x, we can use a method called elimination. The elimination method works well here because the terms involving 'y' in both equations (\(+\frac{y}{2}\) and \(-\frac{y}{2}\)) have the same coefficient but opposite signs. This means that if we add the two equations together, the 'y' terms will cancel out, or be "eliminated".

Step-by-Step Calculation to Find x

Let's add Equation 1 and Equation 2:

Equation 1: \(\frac{2x}{3} + \frac{y}{2} = 4\)

Equation 2: \(\frac{x}{3} - \frac{y}{2} = 1\)

Adding the corresponding sides of the equations:

\((\frac{2x}{3} + \frac{y}{2}) + (\frac{x}{3} - \frac{y}{2}) = 4 + 1\)

Group the 'x' terms and 'y' terms:

\((\frac{2x}{3} + \frac{x}{3}) + (\frac{y}{2} - \frac{y}{2}) = 5\)

Combine the terms. The 'y' terms cancel out:

\(\frac{2x + x}{3} + 0 = 5\)

\(\frac{3x}{3} = 5\)

Simplify the equation:

\(x = 5\)

Result: The Value of x

By applying the elimination method to the given system of linear equations, we successfully isolated the variable x and found its value.

The value of x is 5.

Revision Table: Steps for Solving Systems by Elimination

Additional Information: Different Methods for Solving Systems

While the elimination method was very efficient for this particular system, there are other common methods used to solve systems of linear equations:

• Substitution Method: Involves solving one equation for one variable (like solving for x in terms of y), and then substituting that expression into the other equation.
• Graphing Method: Involves plotting both equations on a coordinate plane. The point where the lines intersect is the solution to the system. This method can sometimes be less precise if the intersection point is not exactly on grid lines.
• Matrix Method: More advanced methods using matrices can also be used, especially for larger systems with many variables.

Each method has its advantages depending on the form of the equations in the system.