4.3 Application problems dealing with systems of linear equations

UP and DOWN the River A boat averages 35 miles per hour with the current and 28 miles per hour against the current. What would the boats speed be in still water? I know you can figure it out, but let’s practice the algebra!

UP and DOWN the River Let s = the speed of the boat Let c = the current speed Speed of boat with currents + c = 35 Speed of boat against the currents – c = 28 Add the two equation together and solve for s.

UP and DOWN the River After adding you have Thus, s = 31.5 mph

Two Angles Problem If two angles are complementary (sum = 90 degrees), and the larger angle is 20 less than 3 times the smaller angle, find the two angles.

Two Angles Problem Let x = the small angle Let y = the larger angle Also, y = 3x -20

Two Angles Problem

Use substitution to solve Thus,

Two Angles Problem Now solve this for x, and then find both of the angles

Two Angles Problem Is the total 90?

$$ SALARY not Celery $$ John, a car sales man, earns a weekly rate plus commission on sales all his sales. In week one his salary was $1000 on $40,000 sales. The next week his salary was $1120 on $52,000 sales. What is John’s commission rate and weekly salary?

$$ SALARY not Celery $$ Salary = weekly pay + (sales)(commission %) Let x = the weekly pay Let y = the commission Week 1: 1000 = x + (40000)(y) Week 2: 1120 = x + (52000)(y)

$$ SALARY not Celery $$ Week 1: 1000 = x + (40000)(y) Week 2: 1120 = x + (52000)(y) Solve using the addition method!

$$ SALARY not Celery $$ Week 1: 1000 = x + (40000)(y) Week 2: 1120 = x + (52000)(y) Multiply Week 2 by (-1) Thus, Week 1: 1000 = x y Week 2: = -x y Now add the two equation together, and…..

$$ SALARY not Celery $$ Solve for y and remember what y represents, then use substitution to solve for x.

$$ SALARY not Celery $$ So, the commission rate is 1% and Week 1: 1000 = x (1%) Solve for x (weekly pay).

$$ SALARY not Celery $$ Week 1: 1000 = x (1%) Solve for x (weekly pay). X = 600 (weekly pay)

Solution mixture problem A weed killer with an active ingredient of 24% is to be mixed with water (0%) to make a 100 gallon mixture with 18% active ingredient. How much of each should be added?

Solution mixture problem Let x = Gallons of Weed killer (24%) Let y = Gallons of Water (0%) Why.18 here?

Solution mixture problem Solve #2 for x and then use #1 to find y #1 #2

Solution mixture problem Therefore, 75 gallons of weed killer should be mixed with 25 gallons of water to make 100 gallon mixture with 18% active ingredient.