Question

How many solutions does this equation have?

-p-9=-6-p
no solution
one solution
infinitely many solutions?

Answer 1

Let's solve the equation and determine the number of solutions:

-p - 9 = -6 - p

First, notice that we have "-p" terms on both sides of the equation. When we subtract "-p" from both sides, it cancels out:

-9 = -6

Now, let's analyze the resulting equation: -9 = -6. This equation is not true; -9 is not equal to -6.

Since this equation leads to a contradiction (a statement that is always false), it has no solution.

Answer 2

usually whenever you have a system of solutions and the result either by substitution or elimination ends up like something 0 = 25, or 15 = -3 or something absurd like that, is another way of saying no solutions.

Why's that?

well, notice one equation say -p-9 has a slope of -1, and the other has -6-p also a slope of -1, same slope = parallel lines, they're parallel but with a different y-intercept since they're in slope-intercept form, that means, two parallel lines apart from each other, well, since they're parallel they will never intersect, and solutions to the system is where they intersect. No intersections, No solutions.