Thursday, July 16, 2020

Linear Equations Review - Word Problems - Notes

Here are the notes for Linear Equations Review - Word Problems Section in the Linear Programming Unit.  Let me know if you have any questions or need any further assistance.

Example: You are going to rent a car for a day. You have two choices, Speed Car Rental and Honest Car Rental. Speed charges $18 plus $0.85 per mile while Honest charges $42 plus $0.45 per mile. a. Develop an equation for the cost of renting a car from Speed. The equation for Speed is c = 0.85m + 18. b. Develop an equation for the cost of renting a car from Honest. The equation for Honest is c = 0.45m + 42. c. Find the intersection of the two lines. Label the point. Note that 0.85m + 18 = 0.45m + 42 implies that 0.40m + 18 = 42 implies that 0.40m = 24 implies that m = 60. Put m = 60 into one of 0.85m + 18 or 0.45m + 42. We put m = 60 into 0.85m + 18 to get 0.85(60) + 18 = 51 + 18 = 69.

The lines intersect at (m, c) = (60, 69). d. Graph both equations on the same set of axes. Label each axis and choose an appropriate scale. Only graph the portion that is relevant to the problem. Note that m (number of miles) is the input (independent variable) and will be put on the "x-axis". Note that c (cost) is the output (dependent variable (because c depends on m)) and will be put on the y-axis. Label the c-intercepts (cost-intercepts) which occur when m = 0. Label the point of intersection of both lines, the origin, and the equations of both of the lines.

e. Use the graph to determine when Speed costs more than Honest. Speed is more expensive than honest when m > 60. f. Use the graph to determine when Honest costs more than Speed. Honest is more expensive than Speed when the graph of Honest is above the graph of Speed. This occurs when m is less than 60 and also greater than or equal to 0. g. What do the cost-intercepts mean in terms of the problems? The cost intercept of Speed is where the line for Speed crosses the cost axis (when m = 0). This is the point (m, c) = (0, 18). The cost of going zero miles is $18. The cost intercept of Honest is where the line for Honest crosses the cost axis (when m = 0). This is the point (m, c) = (0, 42). The cost of going zero miles is $42. h. What does the slope of each line mean in terms of the problem? Aside: We can find the slope of each line by using the slope-intercept equation y = mx + b. The number multiplying x (the coefficient of x, which is m) is the slope of the line. For Speed, c = 0.85m + 18, the slope of the line is 0.85. For example, for Honest, c = 0.40m + 0.42, In both cases, the slope of the line is the cost per mile and indicates that steepness of the line, and the rate of cost is increasing per mile.

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