6-5 Applying Systems

9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.

California Standards

6-5 Applying Systems

When a kayaker paddles downstream, the river’s current helps the kayaker move faster, so the speed of the current is added to the kayaker’s speed in still water to find the total speed. When a kayaker is going upstream, the speed of the current is subtracted from the kayaker’s speed in still water.You can use these ideas and a system of equations to solve problems about rates of speed.

6-5 Applying Systems

Additional Example 1: Solving Rate ProblemsWith a tailwind, an airplane makes a 900-mile trip in 2.25 hours. On the return trip, the plane flies against the wind and makes the trip in 3 hours. What is the plane’s speed? What is the wind’s speed?

Let p be the rate at which the plane flies in still air, and let w be the rate of the wind.

Use a table to set up two equations–one for against the wind and one for with the wind.

6-5 Applying Systems

rate time = distanceRemember!

6-5 Applying Systems

Rate Time = Distance

Upwind p – w 3 = 900Downwind p + w 2.25 = 900

Solve the system 3(p – w) = 900 2.25(p + w) = 900.

First write the system as 3p – 3w = 900 2.25p + 2.25w = 900,

and then use elimination.

Additional Example 1 Continued

6-5 Applying SystemsCheck It Out! Example 1

Ben paddles his kayak along a course on a different river. Going upstream, it takes him 6 hours to complete the course. Going downstream, it takes him 2 hours to complete the same course. What is the rate of the current and how long is the course?

Let d be the distance traveled, and let c be the rate of the current.

Use a table to set up two equations–one for the upstream trip and one for the downstream trip.

6-5 Applying Systems

Rate Time = Distance

Upstream 3 – c 6 = d

Downstream 3 + c 2 = d

Solve the system 6(3 – c) = d 2(3 + c) = d.

First write the system as 18 – 6c = d 6 + 2c = d,

and then use elimination.

Check It Out! Example 1 Continued

6-5 Applying SystemsAdditional Example 2: Solving Mixture ProblemsA chemist mixes a 20% saline solution and a 40% saline solution to get 60 milliliters of a 25% saline solution. How many milliliters of each saline solution should the chemist use in the mixture?

Let t be the milliliters of 20% saline solution and f be the milliliters of 40% saline solution.

Use a table to set up two equations–one for the amount of solution and one for the amount of saline.

6-5 Applying SystemsAdditional Example 2 Continued

20% + 40% = 25%

Solution t + f = 60Saline 0.20t + 0.40f = 0.25(60) = 15

Solve the system t + f = 60 0.20t + 0.40f = 15.

Use substitution.

Saline Saline Saline

6-5 Applying SystemsCheck It Out! Example 2

Suppose a pharmacist wants to get 30 g of an ointment that is 10% zinc oxide by mixing an ointment that is 9% zinc oxide with an ointment that is 15% zinc oxide. How many grams of each ointment should the pharmacist mix together?

9% Ointment + 15%

Ointment = 10% Ointment

Ointment (g) s + t = 30Zinc Oxide (g) 0.09s + 0.15t = 0.10(30) = 3

Solve the system s + t = 30 0.09s + 0.15t = 3.

Use substitution.

6-5 Applying SystemsAdditional Example 3: Solving Number-Digit ProblemsThe sum of the digits of a two-digit number is 10. When the digits are reversed, the new number is 54 more than the original number. What is the original number?

Let t represent the tens digit of the original number and let u represent the units digit. Write the original number and the new number in expanded form.Original number: 10t + uNew number: 10u + t

6-5 Applying SystemsCheck It Out! Example 3

The sum of the digits of a two-digit number is 17. When the digits are reversed, the new number is 9 more than the original number. What is the original number?

Let t represent the tens digit of the original number and let u represent the units digit. Write the original number and the new number in expanded form.Original number: 10t + uNew number: 10u + t

6-5 Applying SystemsLesson Quiz: Part I

1. Allyson paddles her canoe 9 miles upstream in 4.5 hours. The return trip downstream takes her 1.5 hours. What is the rate at which Allyson paddles in still water? What is the rate of the current? 4 mi/h, 2mi/h

2. A pharmacist mixes Lotion A, which is 5% alcohol, with Lotion B, which is 10% alcohol, to make 50 mL of a new lotion that is 8% alcohol. How many milliliters of Lotions A and B go into the mixture? 20 mL of Lotion A and 30 mL of Lotion B.

6-5 Applying SystemsLesson Quiz: Part II

3. The sum of the digits of a two digit number is 13. When the digits are reversed, the new number is 9 less than the original number. What is the original number? 76