Question

1. the amount of money (y) that a salesperson makes is based on the number of items sold (x). the salesperson makes $31,500 if she sells 10 items and $33,750 if she sells 25 items. she connects these two points with a straight line to estimate her future sales. write an equation in point - slope form that models the situation.
2. is $6x + 2y = 24$ equivalent to $y = 3x + 12$? why or why not?

Explanation:

Question 18

Step1: Find the slope

We have two points \((x_1,y_1)=(10, 31500)\) and \((x_2,y_2)=(25, 33750)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
So \(m=\frac{33750 - 31500}{25 - 10}=\frac{2250}{15} = 150\).

Step2: Use point - slope form

The point - slope form of a line is \(y - y_1=m(x - x_1)\). Let's use the point \((10,31500)\) and \(m = 150\).
Substitute into the formula: \(y-31500 = 150(x - 10)\).

Step1: Solve \(6x + 2y=24\) for \(y\)

First, subtract \(6x\) from both sides: \(2y=-6x + 24\).
Then, divide each term by 2: \(y=\frac{-6x + 24}{2}=-3x + 12\).

Step2: Compare with \(y = 3x+12\)

The equation \(6x + 2y = 24\) simplifies to \(y=-3x + 12\), and the other equation is \(y = 3x+12\). The slopes are \(- 3\) and \(3\) (different) and the y - intercepts are both 12 (same). Since the slopes are different, the lines are not the same, so the equations are not equivalent.

Answer:

The equation in point - slope form is \(y - 31500=150(x - 10)\)

Question 19