Algebraic question. Linearity

A magazine company made a yearly profit P of $98 000 when the number of subscribers S totalled 32 000. When the number of subscribers increased to 35 000, the yearly profit was calculated to be $117 500. Assuming the profit P is a linear function of the number of subscribers S:

Write the profit function P(s).
What would the predicted yearly profit be when the number of subscribers reaches 50 000?

the given data can be expressed as coordinates of a point:

(s,P)

the coordinates of the first point would be

(s1,P1) is (32,98)

and the coordinates of the second point would be

(s2,P2) is (35,117.5)

using the slope-intercept form of equation of a line we can formulate two equations from the given coordinates

y = mx + b -->slope intercept form

for the first point (s1,P1) is (32,98)

98 = m*(32) + b --->(eq.1)

for the second point (s2,P2) is (35,117.5)

117.5 = m*(98) + b --->(eq.2)

we now have two equations with two unknown variables. solve for the variables by simultaneous equation;

( 98.0 = m*(32) + b )*-1
( 117.5= m*(35) + b)


( -98.0 = -m*(32) - b)
( 117.5 = m*(35) + b)

adding the two equation will eliminate the variable b and allows us to solve for m

-98.0 + 117.5 = -m*32 + m*35 - b + b
19.5 = m*3
(19.5)/3 = m

plug in the value of m to either of eq.1 or eq.2 (i will use eq.1)

98 = m*(32) + b
98 = (19.5/3)*(32) + b
98 = (624/3) + b
98 - (624/3) = b
(-330/3) = b
-110 = b

thus.the required profit function

P(s) = (19.5/3)*s - 110 -->(eq.3)

to predict the yearly profit when the number of subscribers reaches 50000 use eq.3

P(s) = (19.5/3)*s - 110
P(50) = (19.5/3)*(50) - 110
P(50) = (975/3) - 110

P(50) = 325 - 110
P(50) = 215 thousand or 215,0000 <-----answer

P = AS +B
98000 =32000A +B.............(i)
117500 =35000A + B ..........(ii)
by subtracting
19500 =3000A
A = 19500/3000 =195/30 = 6.5.......(iii)
From (i)
98000 =32000*6.5 + B
98,000 =211,250+B
B = -211,250 +98,000
B =-113,250.....................(iv)
P =(6.5)S -113,250 ...............Ans

P(s)=6.5s-110,000