Question 714737
Let x = gap width in inches and y = temperature in Fahrenheit




"Suppose that a bridge's expansion joint has a gap of 1/2 inches when the temperature if 75°F" means x = 1/2 and y = 75




"and that gap increases to 1 inch when the temperature is 50°F " means x = 1 and y = 50


Let's find the slope of the line through these two points


Slope: 


m = (y2 - y1)/(x2 - x1)


m = (50 - 75)/(1 - 1/2)


m = (-25)/(1/2)


m = -50


So the slope is m = -50


Now use this to find the equation 


y = mx + b


y = -50x + b ... plug in the slope we just found


50 = -50(1) + b ... plug in x = 1 and y = 50


50 = -50 + b


50 + 50 = b


100 = b


b = 100 


So the equation is y = -50x + 100 


Replace x with W since x = gap and y with T because y = temp to get


T = -50W + 100


Now solve for W (because we want an equation that expresses the width of the gap (W) as a function of the temperature (T))


{{{T = -50W + 100}}}


{{{T-100 = -50W}}}


{{{-50W = T-100}}}


{{{W = (T-100)/(-50)}}}


{{{W = -(T/50)+100/(50)}}}


{{{W = -T/50+2}}}



So the final answer is {{{W = -T/50+2}}}