So far we have looked at linear systems of equations in which the lines always intersected in one, unique point. ... When we graph them, they are one line, coincident, meaning they have all points in common. This means that there are an infinite number of solutions to the system.
Exactly one solution: For an equation to have exactly one solution, you must be able to solve for x. A, C, and D all have 30x in their equation. Meaning, if you equated them, you would either end up with a false statement or a true statement.
Therefore, we have two possible equations that has one solution.
30x + 5 = 32.5x 30x + 10 = 32.5x
No solution:
For an equation to have no solutions, the equation, when simplified fully, must be a false statement. Combining A and C will result in an equation with infinite solutions( will discuss later)
However, both A and D have the same coefficient but different integers.
30x + 5 = 30x + 10
When simplified, you result with 5 = 10, which is false. Any equation that simplifies to a false statement has no solutions.
Infinite solutions: an equation with an infinite set of solutions will simplify to a true statement. We notice that C and A, after simplification, are the same.
30x + 5 = 30x + 5
After simplification, 5 = 5, which is true. You can try plugging in any value for x and it will always be equal.
