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1.3 Linear Equations in Two Variables - Writing equations from data/points
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An economist modeled the demand Q for a certain product as a linear function of the selling price P. The demand was 20,000 units when the selling price was $40 per unit, and the demand was 15,000 units when the selling price was $60 per unit. Based on the model, what is the demand, in units, when the selling price is $55 per unit?

A.

16,250

B.

16,500

C.

16,750

D.

17,500

An economist modeled the demand Q for a certain product as a linear function of the

Hard-difficulty · SAT Math · Linear Equations in Two Variables — Writing equations from data/points. Read the question above, select your answer, and check the full explanation below to understand exactly why the correct choice works.

Answer explanation

Choice A is correct. Let the economist’s model be the linear function Q equals, m P plus b, where Q is the demand, P is the selling price, m is the slope of the line, and b is the y-coordinate of the y-intercept of the line in the xy-plane, where y equals Q. Two pairs of the selling price P and the demand Q are given. Using the coordinate pairs P comma Q, two points that satisfy the function are 40 comma 20,000 and 60 comma 15,000. The slope m of the function can be found using the formula m equals, the fraction with numerator Q sub 2, minus Q sub 1, and denominator P sub 2, minus P sub 1, end fraction. Substituting the given values into this formula yields m equals, the fraction with numerator 15,000 minus 20,000, and denominator 60 minus 40, end fraction, or  m equals, negative 250. Therefore, Q equals, negative 250 P plus b. The value of b can be found by substituting one of the points into the function. Substituting the values of P and Q from the point with coordinates 40 comma 20,000 yields 20,000 equals, negative 250 times 40, plus b, or 20,000 equals, negative 10,000 plus b. Adding 10,000 to both sides of this equation yields b equals 30,000 . Therefore, the linear function the economist used as the model is Q equals, negative 250 P plus 30,000. Substituting 55 for P yields Q equals, negative 250 times 55, plus 30,000, equals 16,250. It follows that when the selling price is $55 per unit, the demand is 16,250 units.

Choices B, C, and D are incorrect and may result from calculation or conceptual errors.

 

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