G1.03: Examples 4-7

Example 4. A manager is considering the cost C of printing a book based on the number of pages p. He is told that the formula for predicting the cost is linear based on the number of pages and that the y-intercept is $4.50 and the slope is $0.027. Find the formula to predict the cost from the number of pages.

Example 5. Find the formula for the line with slope 1.35 which has the point (5,40) on it.

We can use this same idea to compute the formula for a line if we have two points on it because we can first use the two points to find the slope. Here’s an outline:

Find the formula for of a line through two points.

  1. Choose the appropriate variable to be the output variable and call it y. Then call the input variable x.
  2. Write two points as [latex]({{x}_{1}},{{y}_{1}})[/latex]and [latex]({{x}_{2}},{{y}_{2}})[/latex].
  3. Use the two points to compute the slope. Call it m.
  4. Pick one of the points (either is fine) and call it [latex]({{x}_{0}},{{y}_{0}})[/latex].
  5. Plug those values into this equation. [latex](y-{{y}_{0}})=m(x-{{x}_{0}})[/latex]
  6. Solve for y. That gives the equation of the line.
  7. If different letters are needed besides x and y for the input and output variables, replace the x and y in the formula with those different letters.

Example 6:   Find the formula for the line through (2,6) and (4,11). Identify the slope and y-intercept.

Example 7. We have been told that the amount of oatmeal needed for oatmeal cookies is linearly related to the amount of flour needed. Also, we know that if we use 3 cups of flour, we need 2 cups of oatmeal. And, of course, if we use 0 cups of flour, we will use 0 cups of oatmeal.

  1. Find the formula to predict the oatmeal needed (called M) from the flour needed (F.)
  2. Interpret the slope.
  3. Interpret the y-intercept.

This last example illustrates that sometimes the intercept is not a number that would be realistic in the situation that the problem describes. But it does have a meaning in the algebraic formula.

For a linear formula, the slope is always a number that is meaningful.