**Example 1**: During the first and second quarters of a year, a business had sales of $42,000 and $58,000, respectively. If the growth of sales follows a linear pattern for the next four years, what will sales be in the fourth quarter? In the 9th quarter? Use an algebraic method of solution.

**Solution**:

**1. What are the two variables and their units? **

Ans. Quarter and sales. Quarter is in numbers and sales are in dollars.

**2. What will we predict? (Make it y.) What are the limits? What are some points?**

Ans. We will predict sales. So *y* = sales. Then *x* = quarter and *x* will go from 1 to 16, in increments of 1.

Quarter 1 has sales of 42,000. So x = 1 and y = 42000. Use the point (1, 42000)

Quarter 2 has sales of 58,000. So x = 2 and y = 58000. Use the point (2, 58000)

**3. Is a linear model appropriate?**

We are told to use a linear model here to see what it would predict.

**4. Slope: Find the slope: m = rise / run**

Let (1,42000) be the first point and (2, 58000) be the second point.

[latex]m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{58000-42000}{2-1}=\frac{16000}{1}=16000[/latex]

**Interpret the slope, using the units of the numbers in the problem:**

(As x increases by 1, y increases by m.)

For each quarter that goes by, sales will increase by $16,000.

**5. Write the formula of the linear relationship:**

(Choose either point and use the point-slope form of the line. Then simplify it to the slope-intercept form of the line.)

Choose (1,42000) for the point and use the slope of 16000 that was just computed.

[latex]\begin{align}&\text{}y-{{y}_{0}}=m(x-{{x}_{0}})\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y-42000=16000(x-1)\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y-42000=16000x-16000\\&y-42000+42000=16000x-16000+42000\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y=16000x+26000\\\end{align}[/latex]

**6. Interpret the y-intercept.**

*(The value of y is b when .)*

For quarter 0, the sales would be $26,000. This isn’t meaningful because the quarters start with the first quarter in this problem, not the zero-th quarter.

**7. Use the formula to make the requested prediction. Write the result in a sentence, with units.**

For the fourth quarter, [latex]x=4[/latex].

[latex]\begin{align}&y=16000x+26000\\&y=16000(4)+26000\\&y=64000+26000\\&y=90000\end{align}[/latex]

So, in the fourth quarter, the linear model predicts sales of $90,000.

In a similar manner, for the 9th quarter, the linear model predicts sales of $170,000, because y = 16000(9)+26000 = 170000** **

**8. Sketch a graph from the original problem information, use it to estimate the answers to the prediction questions, and determine whether your answers from the algebraic formula are reasonable.**** **

quarter | sales |

1 | 42000 |

2 | 58000 |

By hand, we plot these points, draw the line, and then extend it far enough to use it to estimate the answers to the questions.

On this graph, we see that, when quarters = 4, the y-value is 90,000.

We also see that when quarters = 9, the y-value is $170,000.

So the answers from the graph are consistent with the answers we by doing the algebra and arithmetic using the formula.

These two answers for the predictions are reasonable.