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 SystemsAdditional Example 2 Continued

Step 1 Solve the first equation for t by subtracting f from both sides.

t + f = 60– f – ft = 60 – f

Step 2 Substitute 60 – f for t in the second equation.

0.20t + 0.40f = 15

0.20(60 – f) + 0.40f = 15

Distribute 0.20 to the expression in parentheses.

0.20(60) – 0.20f + 0.40f = 15

6-5 Applying SystemsAdditional Example 2 Continued

12 – 0.20f + 0.40f = 15Step 3Simplify. Solve for f. 12 + 0.20f = 15

– 12 – 12 0.20f = 3 Subtract 12 from both

sides.

Divide both sides by 0.20.

f = 15

6-5 Applying SystemsAdditional Example 2 Continued

Step 4 Write one of the original equations.

t + f = 60

Substitute 15 for f.t + 15 = 60Subtract 15 from both

sides.

–15 –15 t = 45

Step 5 Write the solution as an ordered pair.

(15, 45)

The chemist should use 15 milliliters of the 40% saline solution and 45 milliliters of the 20% saline solution.

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 Systems

Step 1 Solve the first equation for s by subtracting t from both sides.

s + t = 30– t – ts = 30 – t

Check It Out! Example 2 Continued

Step 2 Substitute 30 – t for s in the second equation.

0.09(30 – t)+ 0.15t = 3

Distribute 0.09 to the expression in parentheses.

0.09(30) – 0.09t + 0.15t = 3

0.09s + 0.15t = 3

6-5 Applying SystemsCheck It Out! Example 2 Continued

Step 3Simplify. Solve for t. 2.7 + 0.06t = 3

– 2.7 – 2.7 0.06t = 0.3 Subtract 2.7 from both

sides.

Divide both sides by 0.06.

t = 5

2.7 – 0.09t + 0.15t = 3

6-5 Applying SystemsCheck It Out! Example 2 Continued

Step 4 Write one of the original equations.

s + t = 30

Substitute 5 for t.s + 5 = 30Subtract 5 from both

sides.

–5 –5 s = 25

Step 5 Write the solution as an ordered pair.

(25, 5)

The pharmacist should use 5 grams of the 15% ointment and 25 grams of the 9% ointment.

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 SystemsAdditional Example 3 Continued

Now set up two equations.The sum of the digits in the original number is 10.

First equation: t + u = 10The new number is 54 more than the original number.

Second equation: 10u + t = (10t + u) + 54

Simplify the second equation, so that the variables are only on the left side.

6-5 Applying SystemsAdditional Example 3 Continued

10u + t = 10t + u + 54 Subtract u from both sides.– u – u9u + t = 10t + 54

Subtract 10t from both sides.

– 10t =–10t 9u – 9t = 54

Divide both sides by 9.

u – t = 6 –t + u = 6

Write the left side with the variable t first.

6-5 Applying SystemsAdditional Example 3 Continued

Now solve the system t + u = 10 –t + u = 6. Use elimination.

Step 1–t + u = + 6

t + u = 10

2u = 16

Add the equations to eliminate the t term.

Step 2 Divide both sides by 2.

u = 8Step 3 Write one of the original

equations.Substitute 8 for u.

t + u = 10 t + 8 = 10

Subtract 8 from both sides.– 8 – 8

t = 2

6-5 Applying SystemsAdditional Example 3 Continued

Step 4 Write the solution as an ordered pair.

(2, 8)

The original number is 28.

Check Check the solution using the original problem.The sum of the digits is 2 + 8 = 10.When the digits are reversed, the new number is 82 and 82 – 54 = 28.

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 SystemsCheck It Out! Example 3 Continued

Now set up two equations.The sum of the digits in the original number is 17.

First equation: t + u = 17The new number is 9 more than the original number.

Second equation: 10u + t = (10t + u) + 9

Simplify the second equation, so that the variables are only on the left side.

6-5 Applying SystemsCheck It Out! Example 3 Continued

10u + t = 10t + u + 9 Subtract u from both sides.– u – u9u + t = 10t + 9

Subtract 10t from both sides.

– 10t =–10t 9u – 9t = 9

Divide both sides by 9.

u – t = 1 –t + u = 1

Write the left side with the variable t first.

6-5 Applying SystemsCheck It Out! Example 3 Continued

Now solve the system t + u = 17 –t + u = 1. Use elimination.

Step 1–t + u = + 1

t + u = 17

2u = 18

Add the equations to eliminate the t term.

Step 2 Divide both sides by 2.

u = 9Step 3 Write one of the original

equations.Substitute 9 for u.

t + u = 17 t + 9 = 17

Subtract 9 from both sides.– 9 – 9

t = 8

6-5 Applying SystemsCheck It Out! Example 3 Continued

Step 4 Write the solution as an ordered pair.

(9, 8)

The original number is 98.

Check Check the solution using the original problem.The sum of the digits is 9 + 8 = 17.When the digits are reversed, the new number is 89 and 89 + 9 = 98.