A manufacturing company finds that they can sell 300 items at $3.50 per item and 500 items at $ 1.50 $1.50 per item. If the relationship between the number of items sold x x and the price per item p p is a linear one:

Find a formula that gives x in terms of p(x)=
Now use the formula to find the number of items they will if the price per item is $2.50.

To find a formula that gives the relationship between the number of items sold (x) and the price per item (p), we can use the information provided:

Let's use the point-slope form of a linear equation:

p - p₁ = m(x - x₁),

where p₁ and x₁ are the coordinates of a point on the line, m is the slope of the line, and p and x are the variables.

We are given two points: (300, $3.50) and (500, $1.50). We can use these points to find the slope (m):

m = (p₂ - p₁) / (x₂ - x₁)
= ($1.50 - $3.50) / (500 - 300)
= -$2 / 200
= -0.01

Now that we have the slope, we can choose any point and substitute it into the point-slope equation to find the formula for x in terms of p.

Let's use the point (300, $3.50):

$3.50 - $3.50 = -0.01(x - 300)
0 = -0.01x + 3

Simplifying the equation, we get:

0.01x = 3
x = 300

Therefore, the formula for x in terms of p is:
x = -0.01p + 300.

To find the number of items they would sell if the price per item is $2.50, we substitute p = $2.50 into the formula:

x = -0.01($2.50) + 300
x = -0.025 + 300
x ≈ 299.975.

Rounding to the nearest whole number, the number of items they will sell if the price per item is $2.50 is approximately 300.

To find a formula that gives the number of items sold (x) in terms of the price per item (p), we can use the concept of linear equations.

Given that the relationship between x and p is linear, we can assume it follows the equation of a line: y = mx + b, where y represents the price per item and x represents the number of items sold.

Let's assign the price per item to the y-axis and the number of items sold to the x-axis. We have two points: (300, 3.50) and (500, 1.50), which gives us two equations:

1. 3.50 = m * 300 + b
2. 1.50 = m * 500 + b

We can solve this system of equations to find the values of m and b.

First, let's subtract equation 2 from equation 1:

3.50 - 1.50 = m * 300 + b - m * 500 - b

Simplifying:

2 = m * 300 - m * 500

Factoring out m:

2 = m * (300 - 500)

Simplifying further:

2 = m * (-200)

Dividing both sides by -200:

m = -2/200
m = -0.01

Now, substitute the value of m into equation 1:

3.50 = -0.01 * 300 + b

Simplifying:

3.50 = -3 + b

Adding 3 to both sides:

6.50 = b

So, we have the values m = -0.01 and b = 6.50. Therefore, the formula that gives the number of items sold (x) in terms of the price per item (p) is:

x = -0.01p + 6.50

To find the number of items they will sell if the price per item is $2.50, substitute p = 2.50 into the formula:

x = -0.01 * 2.50 + 6.50

Simplifying:

x = -0.025 + 6.50

x ≈ 6.475

Therefore, if the price per item is $2.50, the company will sell approximately 6.475 items.

You have two ordered pairs, (300 , 3.5) and (500 , 1.5)

find the slope m and use the equation,
p = mx + b