(05.03 MC)
Determine which system will produce infinitely many solutions.

To determine which system will produce infinitely many solutions, we’re looking for systems whose equations are dependent, meaning they represent the same line.

In the first system, and are parallel lines (different constants), thus no solutions. The second set also has different slopes, resulting in one solution only. However, in the third set, the equations can be manipulated to show they represent the same line, leading to infinitely many solutions.

Therefore, it’s the last system that produces infinitely many solutions! 🎉

Ever heard of parallel lines never meeting? In geometry, it’s a classic illustration of systems that don’t intersect. A fun fact is that lines can be written in various forms (slope-intercept, standard, etc.), but if they don’t have the same slope, you won’t find a point where they cross! Just like those knocking at each other’s doors but never coming in.

When tackling systems of equations, always check for coefficients first! If the ratios of the coefficients of and are equal but the constants are not, you’ve got parallel lines. Also, beware of mixing up signs; for instance, solving the equation incorrectly can lead to believing that solutions are present when they are not. Remember, a simple error is all it takes to draw a line in the wrong direction!