Question

Mario's company made $4 million in 2012. In 2016 they made $13.2 million. a. Write an equation for the profit, y, in term of the x, the years since 2010.$4, 2012

Answer 1

The form of the linear equation is

`y=mx+b`

m is the slope that represents the rate of profit per year

b is the y-intercept which represents the initial amount in the year 2010

Since y is the profit in millions

Since x is the number of the year starting from 2010, then

At 2012:

The profit is 4 million dollars, y = 4

The year is 2012, x = 2

Substitute them in the rule above

`\begin{gathered} 4=m(2)+b \\ 4=2m+b \\ 2m+b=4\rightarrow(1) \end{gathered}`

At 2016:

The profit is 13.2 million dollars, y = 13.2

The year is 2016, x = 6

Substitute them in the rule above

`\begin{gathered} 13.2=m(6)+b \\ 13.2=6m+b \\ 6m+b=13.2\rightarrow(2) \end{gathered}`

Now, subtract equation (1) from equation (2) to eliminate b

`\begin{gathered} (6m-2m)+(b-b)=(13.2-4) \\ 4m+0=9.2 \\ 4m=9.2 \end{gathered}`

Divide both sides by 4 to find m

`\begin{gathered} (4m)/(4)=(9.2)/(4) \\ m=2.3 \end{gathered}`

Substitute m in equation (1) by 2.3

`\begin{gathered} 2(2.3)+b=4 \\ 4.6+b=4 \end{gathered}`

Subtract 4.6 from both sides

`\begin{gathered} 4.6-4.6+b=4-4.6 \\ b=-0.6 \end{gathered}`

m = 2.3 million dollars per year

b = -0.6 million dollar

The equation of the profit is

`\begin{gathered} y=2.3x+(-0.6) \\ y=2.3x-0.6 \end{gathered}`

The answer is

y = 2.3x - 0.6, where y is the profit in million and x is the years since 2010