Question

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&-4 x+y=-8\\\&-12 x+3 y=-24\end{aligned} $$

Short answer

The system has infinitely many solutions because the two equations are representing the same line.

Step-by-step solution

Checking Consistency

Divide the second equation by \(-3\) to simplify the system.\[-12x+3y=-24 → 4x - y = 8\]. Now it can be seen that it's identical to the first equation \(-4x + y = -8\).

Evaluating the system

Having both equations identical means it's the same line in the coordinate plane. Therefore, every point on this line is a solution to the system.

Counting the Solutions

As the two equations represent the same line, it means this linear system has infinitely many solutions.