91Ó°ÊÓ

The substitution method is a powerful tool when solving systems of equations, as it simplifies calculations by allowing us to deal with one equation at a time. This method involves expressing one variable in terms of the other and then substituting this expression into the second equation.

In our exercise, we first introduced substitution by setting \( a = \frac{1}{x} \) and \( b = \frac{1}{y} \). This clever step converted the complex-looking fractions into a more manageable form. Once simplified, we had equations in terms of \( a \) and \( b \):

• \( a + 2b = \frac{7}{12} \)
• \( 3a - 2b = \frac{5}{12} \)

The substitution method made it possible to focus on finding these specific values rather than worrying about the reciprocals \( x \) and \( y \) immediately. Once we found \( a = \frac{1}{4} \), substituting it back into one of the equations allowed us to find \( b \), and so on until we had our original variables solved.

Linear equations form the core of many mathematical problems, including the exercise at hand. A linear equation is essentially a first-degree polynomial equation in one or more variables. They yield a straight line when graphed, thus the term 'linear.' Understanding them is vital because they provide a foundation for solving real-world problems.

The system we worked with transformed into linear equations by our substitution method:

• \( a + 2b = \frac{7}{12} \)
• \( 3a - 2b = \frac{5}{12} \)

Each equation represents a linear relationship between \( a \) and \( b \). Solving such systems requires finding the point where the equations intersect, which corresponds to the values of \( a \) and \( b \) in this case. This is done through techniques like adding the equations to eliminate variables or graphically finding the intersection point.

Verifying the solution of a system of equations is a crucial step to ensure the result is correct and satisfies the original equations. It acts as a double-check against errors that might occur during calculations.

In the given problem, after solving for \( x \) and \( y \), we proceeded to verify by plugging these variables back into the original equations:

• First equation: \( \frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12} \)
• Second equation: \( \frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \)

Both expressions accurately simplified to satisfy the given equations, confirming that \( x = 4 \) and \( y = 6 \) are indeed the correct solutions. This step reassures us that no mistakes were made and that our solutions are reliable.