Question

write an equation is slope-intercept form for the line that satisfies each set of conditions: slope is 5, passes through the origin

Answer 1

Parent equation for every line. Think of this as a function where we choose a, b, c and then we pass any value of x, to get a y.
y = a(x-b) + c

The above equation is a fancier version of its parent which is y=x, here (a=1,b=0,c=0). The discussion below will refer to the simple parent y=x, which passes through the origin.

Variables:
c is the "vertical shifter" of the line. If a = b = 0, then our equation is y = c. When c = any number the entire function moves up or down. Another way to think about it is to think of the parent as two things a(x-b) and c. No matter what a(x-b) evaluates to c will always add a certain amount to the output. Graphically a(x-b) for all x will equal something line, then we add some c to every point. It makes sense this shifts everything up

a is the "scaler". Basically it scales our input. Look at the function y=ax. Here every value gets passed as x, and the last thing that happens before we get an output is that it gets scaled by a. Also think about y=1*x. Where a = 1, here every one of our inputs gets scaled by one (doesn't change anything) before becoming an output. Ultimately this stretches our function

b is the "horizontal shifter" or input shifter. Whenever we input a value for x, it gets changed by the shifter before its scaled by a, or vertically shifted by c. Its a really quick way to treat an x value as another x. Ex. y=2x, and y=2(x-3). When we feed any number to the first equation its directly scaled i.e y(3) = 6, and 3 corresponds with the output 6. In our second equation and input of 3 really quickly becomes a value of 0 before being scaled. It's almost as if, we can make every input act like the input 3 before. This ultimately causes a horizontal shift.

If a function has a slope of 5, then it has a scaler of 5. If it passes through the origin then its just the parent in terms of the horizontal shifter. Every function which has a 0 for the horizontal shifter, always passes through the origin.