91Ó°ÊÓ

The addition method, also known as the elimination method, is a popular technique used to solve systems of linear equations. The fundamental goal is to eliminate one of the variables by adding or subtracting the equations, making it easier to solve for the remaining variable. In our system, we have equations with variables in different terms. By manipulating the equations to align the coefficients (especially making them opposites), we can eliminate a variable. For example, after clearing fractions, we manipulated our equations so that the coefficients of \( x \) became 12 and -12. Adding these terms effectively cancels out the \( x \), allowing us to focus solely on solving for \( y \) first.

Fractions in systems of equations can make handling calculations more complex and error-prone. To simplify, it's often helpful to clear fractions. This involves multi-step processes of multiplication by the least common multiple (LCM) of all denominators in each equation.
To clear fractions from our system:

• The first equation is multiplied by 4 (LCM of 2 and 4), resulting in: \(12x + 14y = 3\).
• The second equation is multiplied by 6 (LCM of 2, 3, and 4) initially, followed by multiplying by 2 for further simplification, resulting in \(-6x + 20y = -15\).

Why multiply by LCM? It ensures that all fraction denominators are simplified to whole numbers, establishing consistency across the equations. This step helps significantly when aiming for straightforward addition and subtraction, as seen in the solution steps.

Eliminating a variable in systems of equations simplifies solving. Once we achieve similar (or opposite) coefficients of a variable across both equations, addition or subtraction can remove that variable altogether. In this exercise, we focused on eliminating \( x \) to solve for \( y \).
The transformation of our equations to \(12x + 14y = 3\) and \(-12x + 40y = -30\) was a strategic move. Here, multiplying the second equation ensured that adding it to the first cancels out \( x \) (as \(12x - 12x = 0\)), leading us directly to solve \( 54y = -27 \).
This method of elimination is powerful:

• It reduces two-variable systems to one-variable equations.
• It allows for straightforward solution finding by focusing exclusively on one variable at a time.

Verifying solutions is a crucial step to ensure accuracy and attention to detail in solving systems of equations. Once we obtained \( x = \frac{5}{6}\) and \( y = -\frac{1}{2}\), we substituted these values back into the original equations to confirm correctness.
Verification involves:

• Substitute found values into the original equations.
• Check that the equations simplify to true statements. For example:
  • First equation: \(3\left(\frac{5}{6}\right) + \frac{7}{2}\left(-\frac{1}{2}\right) = \frac{3}{4}\), simplifying confirms equality.
  • Second equation: \(-\frac{5}{12} + \frac{5}{3}\left(-\frac{1}{2}\right) = -\frac{5}{4}\), confirming in a similar way.

This process is essential not just for self-checking but also in rigorous settings like exams. It eradicates doubts and guarantees that every step taken is reliable.