Warm Up Simplify each expression. 1. 3(10a + 4) – 2 2. 5(20 – t) + 8t 3. (8m + 2n) – (5m + 3n) 30a + 10 100 + 3t 3m – n 4. y – 2x = 4 x + y = 7 Solve by.

Warm UpSimplify each expression.

1. 3(10a + 4) – 2

2. 5(20 – t) + 8t

3. (8m + 2n) – (5m + 3n)

30a + 10

100 + 3t

3m – n

4. y – 2x = 4x + y = 7

Solve by using any method.

(1, 6) 5. 2x – y = –1y = x + 5

(4, 9)

Lesson 6.4

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

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.

rate time = distanceRemember!

With 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.

Rate Time = Distance

Upwind p – w 3 = 900

Downwind p + w 2.25 = 900

Solve the system3(p – w) = 900

2.25(p + w) = 900.

First write the system 3p – 3w = 900

2.25p + 2.25w = 900,

and then use elimination.

Step 1 –0.75(3p – 3w = 900)

2.25p + 2.25w = 900

Multiply each term in the first equation by –0.75 to get opposite coefficients of p.

Add the new equation to the second equation.

–2.25p + 2.25w = –675

+ 2.25p + 2.25w = 900

4.5w = 225 Simplify and solve for w.

w = 50

Step 2

Write one of the original equations.

Substitute 50 for w.

Step 3 3p – 3w = 900

3p – 3(50) = 900

Add 150 to both sides.3p – 150 = 900

+ 150 + 150

3p = 1050

Divide both sides by 3.

p = 350

The plane’s speed is 350 mi/h and the wind’s speed is 50 mi/h.

Step 4 (350, 50)

Write the solution as an ordered pair.

nickels + dimes = total

Number of coins

Value in dollars

A jar contains a total of 38 nickels and dimes. The total value of the coins is $2.75 How many nickels and how many dimes are in the jar?

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.

20% + 40% = 25%

Solution t + f = 60

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

Solve the systemt + f = 60

0.20t + 0.40f = 15.

Use substitution.

Saline Saline Saline

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

t + f = 60

– f – f

t = 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

12 – 0.20f + 0.40f = 15Step 3

Simplify. Solve for f. 12 + 0.20f = 15

– 12 – 12

0.20f = 3Subtract 12 from both sides.

Divide both sides by 0.20.

f = 15

Step 4 Write one of the original equations.

t + f = 60

Substitute 15 for f.t + 15 = 60

Subtract 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.

Now set up two equations.

The sum of the digits in the original number is 17.

First equation: t + u = 17

The 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.

Solving Number-Digit ProblemsThe 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 + u

New number: 10u + t

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

9u + 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.

Now solve the systemt + 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 = 9

Step 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

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.

Lesson Quiz: Part I1. 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.

Lesson Quiz: Part II3. 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