Questions: Solve the system by the addition (elimination) method: -6x + 2y = -6 12x - 4y = 12

Solution Steps

To solve the system of equations using the addition (elimination) method, we need to eliminate one of the variables by adding the equations together. In this case, we can see that the coefficients of \(x\) and \(y\) are multiples of each other. By adding the two equations directly, we can eliminate both variables and check for consistency.

We start with the given system of equations: \[ \begin{array}{l} -6x + 2y = -6 \\ 12x - 4y = 12 \end{array} \]

We can simplify both equations by dividing them by their respective coefficients: \[ \begin{array}{l} -6x + 2y = -6 \quad \Rightarrow \quad -3x + y = -3 \quad \text{(divide by 2)} \\ 12x - 4y = 12 \quad \Rightarrow \quad 3x - y = 3 \quad \text{(divide by 4)} \end{array} \]

Next, we add the simplified equations to eliminate both variables: \[ \begin{array}{l} -3x + y = -3 \\ 3x - y = 3 \\ \hline 0 = 0 \end{array} \] The result \(0 = 0\) is always true, indicating that the system of equations is dependent and has infinitely many solutions.

Since the system has infinitely many solutions, we can express one variable in terms of the other. From the first simplified equation: \[ y = 3x - 3 \] Thus, the solution can be written as: \[ (x, y) = (x, 3x - 3) \] where \(x\) is any real number.

Final Answer