A municipal water utility charges customers a flat monthly fee plus a constant rate per th… — SAT

A municipal water utility charges customers a flat monthly fee plus a constant rate per thousand gallons of water used. In June, a household used $8$ thousand gallons of water and was charged a total of $\$46.00$ . In July, the same household used $12$ thousand gallons of water and was charged a total of $\$62.00$ . If $x$ represents the number of thousand gallons of water used in a month, and $y$ represents the total monthly charge, in dollars, which of the following equations represents the relationship between $x$ and $y$ ?

A
$y = 0.25x + 44$
B
$y = 14x + 4$
$y = 4x + 14$ Answer
D
$y = 4x + 78$

Answer

y = 4x + 14

The correct equation is

y = 4x + 14

. The constant rate of change (slope) is found by dividing the difference in monthly charges by the difference in water usage:

\frac{62 - 46}{12 - 8} = 4

. Using the point-slope form or slope-intercept form with the point

(8, 46)

yields

46 = 4(8) + b

, which simplifies to

b = 14

for the y-intercept (the flat monthly fee). Substituting these parameters back into the slope-intercept form yields the correct relationship.

Step-by-Step Solution

Calculate the constant rate of change (slope,

m

) using the two given data points,

(8, 46)

and

(12, 62)

m = \frac{62 - 46}{12 - 8} = \frac{16}{4} = 4

The constant rate of change represents the slope of the linear equation.

Use the slope-intercept form,

y = mx + b

, and substitute one of the data points, such as

(8, 46)

, to solve for the y-intercept (

b

46 = 4(8) + b \implies 46 = 32 + b \implies b = 14

Substituting a known point allows us to isolate and find the value of the flat monthly fee, which is the y-intercept.

Write the final equation by substituting the calculated slope (

m = 4

) and y-intercept (

b = 14

) back into the slope-intercept form.

y = 4x + 14

This combines the rate per thousand gallons and the flat fee into the final linear relationship.

Key Concept

Linear Equations in Two Variables

Estimated Time:1m 30s

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