The table of values represents a linear function g(x), where x is the number of days that have passed and g(x) is the balance in the bank account: x g(x) 0 $325 5 $400 10 $475 Part A: Find and interpret the slope of the function. (3 points) Part B: Write the equation of the line in point-slope, slope-intercept, and standard forms. (3 points) Part C: Write the equation of the line using function notation. (2 points) Part D: What is the balance in the bank account after 12 days? (2 points)

Part A)

The slope m is 15.

This means that $15 is deposited into the bank account each day.

Part B)

Point-slope form:

Slope-intercept form:

Standard Form:

Part C)

Part D)

After 12 days, the balance of the bank account will be $505.

Step-by-step explanation:

We are given a table of values that lists x, the number of days that have passed, and g(x), the balance of the bank account.

The table represents a linear function.

Part A)

To find the slope, we can use the slope formula:

Where (x₁, y₁) and (x₂, y₂) are two points.

So, let’s pick two points from our table. We can use (0, 325) and (5, 400).

Let (0, 325) be (x₁, y₁) and let (5, 400) be (x₂, y₂). Substitute the values into the slope formula:

Evaluate:

So, the slope is 15 (or, more precisely, 15/1).

This means that $15 gets deposited into the bank account each day.

Part B)

We want to write the equation of the line in point-slope form, slope-intercept form, and standard form.

A) Point-slope form is given by:

Where m is the slope and (x₁, y₁) is a point.

We will substitute 15 for m.

We can use any point, but let’s use (0, 325) for (x₁, y₁) for consistency. Hence:

And this is point-slope form. You may choose to simplify the zero.  

For reference, here are the other two forms possible given the table:

B) Slope-intercept form is given by:

Where m is the slope and b is the y-intercept.

Remember that the y-intercept is where x=0.

From our table, we can see that when x=0, y=325.

Therefore, our y-intercept is 325.

So, we will substitute 325 for b.

And we will also substitute 15 for m. Therefore:

C) Standard form is given by:

Where A, B, and C are integers, and A is positive.

We can start with the slope-intercept form:

And convert this to standard form. Subtract 15x from both sides:

Since A should be positive, multiply everything by -1. Hence:

Part C)

We can use the slope-intercept form:

Since our dependent variable, y, is the output of our function, y is the same as g(x).

Therefore, we can substitute g(x) for y to acquire:

Part D)

We can use our function. We want to find the balance after 12 days. Hence x=12.

Therefore:

Evaluate:

After 12 days, the balance in the bank account is $505.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4

he has 140 chickens because he has 10 times more chickens than ducks and he has 14 ducks so 14*10=140