Find a Linear Equation

Question

It costs XYZ Corporation $8100 to produce 1400 units and $5580 to produce 200 units. Let c represent the cost in dollars and x represent the number of units produced. Find a linear equation for c in terms of x.

Answer 1

Hi Sally,

 

By linear equation, we mean that we want to find a line that goes through both points.  There are various ways to solve this, but I'm going to recommend we use the two-point form to derive the line.  If you know what I'm talking about, then great.  Otherwise, don't panic, it's not too bad. 

 

The slope of a line is the rise over run between any two points on the line.  For clarity, let's call the slope m.  For example, (1400, 8100) is a point on this line, as is (200, 5580).  And (c, x) is a point on the line too.  So, recall that the slope m of our line is rise over run:

 

m = (8100-5580) / (1400 - 200)

    = (2520) / 1200

    = 2.1

 

Choosing another two points,

 

m = (c - 5580 ) / (x - 200)

 

But if they are both the slope m of our line, they are equal to each other:

 

(c - 5580)

----------------  = 2.1

(x - 200)

 

Multiplying both sides by (x-200), we have

 

c - 5580 = 2.1(x - 200)

 

Add 5580 to both sides, and we have

 

c = 2.1 (x - 200) + 5580

 

And that is a linear equation for c in terms of x.  However, it makes the teacher so much happier if you simplify, so let's do that.

 

c = 2.1 x - 420 + 5580

 

c = 2.1 x + 5160

 

So there's an answer.  But it's important that we check.  We could have made a mistake. 

 

8100 ?=? 2.1 ( 1400) + 5160

8100 ?=? 2940 + 5160

8100 ?=? 8100

:)

 

5580 ?=? 2.1 ( 200) + 5160

5580 ?=? 420 + 5160

5580 ?=? 5580

:)

 

It checks.  Now we're done.