1. The problem asks which method is used to solve a linear system by multiplying one or both equations by appropriate numbers so that one variable has the same coefficient with opposite signs. 2. This method is called the **elimination method** (or addition/subtraction method). 3. The key idea is to manipulate the equations so that adding or subtracting them eliminates one variable, making it easier to solve for the other. 4. For example, given the system: $$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$$ 5. Multiply one or both equations by numbers to get coefficients of $x$ or $y$ that are equal in magnitude but opposite in sign, e.g., $$m \times (a_1x + b_1y) = m c_1$$ $$n \times (a_2x + b_2y) = n c_2$$ where $m a_1 = -n a_2$ or $m b_1 = -n b_2$. 6. Then add the two equations to eliminate that variable: $$ (m a_1 + n a_2) x + (m b_1 + n b_2) y = m c_1 + n c_2 $$ which simplifies to an equation with one variable. 7. Solve for the remaining variable, then substitute back to find the other. This method is very useful when substitution is complicated or when coefficients can be easily manipulated.