78 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.
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Since it is linear, [latex]C=mp+b[/latex], where [latex]m=0.027[/latex] and [latex]b=4.50[/latex].
Thus [latex]C=0.027p+4.50[/latex] is the formula.
Example 5. Find the formula for the line with slope 1.35 which has the point (5,40) on it.
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Let [latex]m=1.35[/latex] and [latex]({{x}_{0}},{{y}_{0}})=(5,40)[/latex]. Then
[latex]\begin{align}&\,\,\,\,\,\,\,\,\,\,\,\,y-{{y}_{0}}=m(x-{{x}_{0}})\\&\,\,\,\,\,\,\,\,\,\,\,\,y-40=1.35(x-5)\\&\,\,\,\,\,\,\,\,\,\,\,\,y-40=1.35x-6.75\\&\,\,y-40+40=1.35x-6.75+40\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y=1.35x+33.25\end{align}[/latex]
Thus the equation of the line is [latex]y=1.35x+33.25[/latex].
Check: We can look at the formula and read the slope as 1.35, so that checks. To see whether the given point is on it, evaluate the formula at [latex]x=5[/latex] and see whether it gives the correct y-value. [latex]y=1.35(5)+33.25=6.75+33.25=40[/latex]. This checks!
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.
- Choose the appropriate variable to be the output variable and call it y. Then call the input variable x.
- Write two points as [latex]({{x}_{1}},{{y}_{1}})[/latex]and [latex]({{x}_{2}},{{y}_{2}})[/latex].
- Use the two points to compute the slope. Call it m.
- Pick one of the points (either is fine) and call it [latex]({{x}_{0}},{{y}_{0}})[/latex].
- Plug those values into this equation. [latex](y-{{y}_{0}})=m(x-{{x}_{0}})[/latex]
- Solve for y. That gives the equation of the line.
- 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.
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Let [latex]({{x}_{1}},{{y}_{1}})[/latex]= (2,6) and = (4,11).
[latex]m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\,\,\,\,\,=\,\,\,\frac{11-6}{4-2}\,\,=\frac{5}{2}\,\,\,\,\,=\,\,\,2.5[/latex]
Now, to use the point-slope form, I will use [latex]({{x}_{0}},{{y}_{0}})[/latex]= (2,6) to get
[latex]\begin{align}&(y-{{y}_{0}})=m(x-{{x}_{0}})\\&(y-6)=2.5(x-2)\\&y-6=2.5x-5\\&y\,\,\,\,\,=2.5x\,\,+\,\,1\\\end{align}[/latex] So the equation of the line is [latex]y=2.5x+1[/latex], the slope is 2.5, the y-intercept is 1.
Check: Use the formula we derived to determine the output value for [latex]x=2[/latex] and see if the given point is on the line. [latex]y=2.5x+1=2.5(2)+1=5+1=6[/latex]. This tells us that (2,6) is on the line.
Use the formula we derived to determine the output value for [latex]x=4[/latex] and see if the second given point is on the line. [latex]y=2.5x+1=2.5(4)+1=10+1=11[/latex]. This tells us that (4,11) is on the line. So this checks.
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.
- Find the formula to predict the oatmeal needed (called M) from the flour needed (F.)
- Interpret the slope.
- Interpret the y-intercept.
Show Answer
a. Here, the output variable (y) is M and the input variable (x) is F.
So the two points given are (0,0) and (3, 2).
Using the slope formula, [latex]m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\,\,\,\,\,=\,\,\,\frac{2-0}{3-0}\,\,=\frac{2}{3}\,\,\,\,\,=\,\,\,0.667[/latex]cups of oatmeal per one cup of flour.
Then, to use the point-slope form of the line, we’ll use and the first point, which is (0,0).
[latex]\begin{align}&(y-{{y}_{0}})=m(x-{{x}_{0}})\\&(y-0)=0.667(x-0)\\&\,\,\,\,\,\,\,\,\,\,\,y=0.667x\\&\,\,\,\,\,\,\,\,\,\,\,y\,=0.667x+0\\\end{align}[/latex]
So the formula is [latex]M=0.667\cdot{F}[/latex], so the slope is 0.667 and the y-intercept is 0.
- Interpret the slope: For every 1 additional cup of flour, we should use 0.667 additional cups of oatmeal.
- Interpret the intercept: If we use 0 cups of flour, then we will use 0 cups of oatmeal.
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.