Exercise 117 Page 264

Both cities population change at a constant rate. This means we can describe the populations with linear functions in slope-intercept form. y= mx+ b In this form, m is the slope and b is the y-intercept. In this case, the y-intercept shows current populations, b= 1532 and b= 2740, for Woottonville and Coynertown, respectively. Woottonville:& y= mx+ 1532 Coynertown:& y= mx+ 2740 We also have to determine the slope. For Woottonville, the population increases by 15 people per year which translates to a slope of m= 15. Conversely, in Coynertown, the population decreases by 32 people per year which can be interpreted as a slope of m= - 32. With this, we can complete the equations. Woottonville:& y= 15x+ 1532 Coynertown:& y= - 32x+ 2740 If we combine the equations, we get a system of equations. y=15x+1532 y=- 32x+2740 By solving this system, we can determine how many years it will take for the population to be the same. Note that both equations are solved for y and therefore, we should use the Substitution Method.

y=15x+1532 & (I) y=- 32x+2740 & (II)

Substitute

(II): y= 15x+1532

y=15x+1532 15x+1532=- 32x+2740

▼

(II): Solve for x

AddEqn

(II): LHS+32x=RHS+32x

y=15x+1532 47x+1532=2740

SubEqn

(II): LHS-1532=RHS-1532

y=15x+1532 47x=1208

DivEqn

(II): .LHS /47.=.RHS /47.

y=15x+1532 x=25.70212...

RoundDec

(II): Round to 1 decimal place(s)

y=15x+1532 x≈ 25.7

After about 25.7 years the cities populations will be the same.

Extra

Why didn't we calculate y?

Note that we are only looking to determine when the populations are the same and not what the populations is at that time. Therefore, we do not have to solve for y.