Matthew K. answered • 12/03/21

High School Algebra Teacher

**Part a:** You can think of g(x) as just being "y" in disguise. If you think about it that way, we have a more familiar "x, y" problem where we can apply the slope and slope-intercept equations.

You can consider the first two lines of your table to be two ordered pairs:

(0, 600) (3, 720)

Then, use the slope equation:

m = (y_{2 }- y_{1})/(x_{2 }- x_{1})

m = (720 - 600)/(3 - 0)

m = 120/3

m = 40

The interpretation they're probably looking for is that this means that $40 is being added to the account every day.

**Part b: **Now usually there would be some additional steps to find the y-intercept "b", but in this case, we got lucky: x_{1} = 0, which means y_{1} is automatically "b".

So now we can write our final equation in **slope-intercept form**:

y = 40x + 600

If you want to check this, just plug in the third "x" value and see what happens:

y = 40(6) + 600

y = 240 + 600

y = 840

That matches the third line of the table, which says you will have $840 after 6 days.

To get to **point-slope form**, just substitute into the point-slope equation:

y - y_{1} = m(x - x_{1})

y - 600 = 40(x - 0)

Your teacher may want you to simplify this to

y - 600 = 40x

**Standard form** is when we write a linear equation in two variables in the form Ax + By = C, with some special rules: A, B, and C must be integers (whole numbers or the negatives of whole numbers), A cannot be negative, and, A, B, and C have no common factors other than 1 (that is, they are *relatively prime*.)

(Note: A, B and C are not *variables* even though they are letters, they are just __unknowns__. They start out as letters because we don't know their values, but they have single values which we discover. In that way, they are like "m" and "b" from slope-intercept form.)

Now, take our simplified point-slope equation as the starting point:

y - 600 = 40x

First, re-arrange so that "x" and "y" are on the same side. Since A, the coefficient of "x", must be positive, we should cancel the "y" term:

y - 600 = 40x

+^{-}1y +^{-}1y

^{-}600 = 40x + ^{-}1y

Now, use the symmetric property (which says that you can "swap" the sides of an equation), to get:

40x + ^{-}1y = ^{-}600.

Let's see if we've followed the rules for standard form:

A is positive - check!

A, B and C are integers - check!

A, B and C are relatively prime - B is ^{-}1, so we can be assured it doesn't have any nontrivial factors in common with A or C. Check!

**Part c: **Let's go back to our slope-intercept form, and remember that g(x) was really just "y" in disguise:

y = 40x + 600

g(x) = 40x + 600

**Part d: **Since days is the unit of "x," substitute 7 into your function for "x":

g(x) = 40x + 600

g(7) = 40(7) + 600

g(7) = 280 + 600

g(7) = 880

So there will be $880 in the account after 7 days.

Another way to think about this question is that we know from the table that we have $840 after 6 days, and we are adding $40 per day to the account, so we will have $840 + $40 = $880 after 7 days. Does all that make sense?