# Maths Question Using Formula for Linear Regression Line

The following question and solution are from Algebra website.

Question

Margarita is hired by an accounting firm at a starting salary of \$58,000 per year. Three years later her annual salary has increased to \$66,400. Assume her salary increases linearly.
a) Write the equation of a linear function that relates her annual salary, S, and the number of years, T, she has worked for the firm.
b) What does the slope of her salary function represent?
c) What does the S-intercept of her salary function represent?
d) Assuming the same rate of growth, after how many years with the firm will her salary first exceed \$100,000? Describe what you did to arrive at your result.

Workings

Starting Salary is \$58,000
She makes \$66,400 in 3 years.
Salary is increasing linearly (in a straight line).
S = Salary
T = number of years.
Her rate of growth is equal to (\$66,400 – \$58,000) / 3 = \$2,800 per year.
Since the annual growth is in the form of a straight line equation, it will take the form of S = m*T + b
m is the slope and b is the S intercept.
The * is the multiplication symbol.
The slope is the change in salary per year which is equal to \$2,800.
Therefore equation becomes:
S = 2800*T + b
The S intercept is the value of S when T = 0.
When T = 0, she was just starting and her salary was \$58,000, so the value of b is equal to \$58,000.
The equation S = m*T + b becomes:
S = 2800*T + 58000

a) S = 2800 * T + 58000
b) The slope of her salary function represents the change in salary each year.
c) The S-intercept of her salary function represents her starting salary.
d) The equation is S = 2800 * T + 58000
Substitute S = \$100,000, into the equation to get:
100000 = 2800 * T + 58000
Subtract 58000 from both sides of this equation to get:
42000 = 2800 * T
Divide both sides of this equation by 2800 to get:
T = 15.
She will reach a salary of \$100,000 per year in the 15th year of her employment.
We will draw a graph this equation for 20 years. In this graph, we replace S with y and T with x.
The equation becomes:
y = 2800 * x + 58000

We drew 3 horizontal lines at y = 58000 and y = 66400 and y = 100000 so you can spot the year in which those values occur easier.
The estimated year in which they occur are:Year 0, Year 3, and Year 15 which can be confirmed easily by replacing x in the equation with 0, 3, and 15, and solving for y.
Note the equation is the same:
S = 2800 * T + 58000 is exactly the same equation as:
y = 2800 * x + 58000.