4.5

Systems of Linear Equations and

Problem Solving

Steps in Solving Problems1) Understand the problem.

• Read and reread the problem.• Choose a variable to represent the unknown.• Construct a drawing, whenever possible.• Propose a solution and check.

2) Translate the problem into two equations.

3) Solve the system of equations.

4) Interpret the results.• Check proposed solution in the problem.• State your conclusion.

Problem Solving Steps

Example

continued

One number is 4 more than twice the second number. Their total is 25. Find the numbers.

Read and reread the problem. Suppose that the second number is 5. Then the first number, which is 4 more than twice the second number, would have to be 14 (4 + 2•5).

Is their total 25? No: 14 + 5 = 19. Our proposed solution is incorrect, but we now have a better understanding of the problem.

Since we are looking for two numbers, we let

x = first number

y = second number

1. UNDERSTAND

continued

2. TRANSLATE

continued

One number is 4 more than twice the second number.

x = 4 + 2y

Their total is 25.

x + y = 25

3. SOLVE

continued

continued

Using the substitution method, we substitute the solution for x from the first equation into the second equation. x + y = 25

(4 + 2y) + y = 25 Replace x with 4 + 2y.

4 + 3y = 25 Simplify.

3y = 21 Subtract 4 from both sides.

y = 7 Divide both sides by 3.

We are solving the system x = 4 + 2yx + y = 25

4. INTERPRET

continued

Check: Substitute x = 18 and y = 7 into both of the equations.

First equation: x = 4 + 2y 18 = 4 + 2(7) True

Second equation: x + y = 25

18 + 7 = 25 True

State: The two numbers are 18 and 7.

Now we substitute 7 for y into the first equation. x = 4 + 2y = 4 + 2(7) = 4 + 14 = 18

Example

continued

Hilton University Drama club sold 311 tickets for a play. Student tickets cost 50 cents each; non-student tickets cost $1.50. If the total receipts were $385.50, find how many tickets of each type were sold.

1. UNDERSTAND

Read and reread the problem. Suppose the number of students tickets was 200. Since the total number of tickets sold was 311, the number of non-student tickets would have to be 111 (311 – 200).

continued

The total receipts are $385.50. Admission for the 200 students will be 200($0.50), or $100. Admission for the 111 non-students will be 111($1.50) = $166.50. This gives total receipts of $100 + $166.50 = $266.50. Our proposed solution is incorrect, but we now have a better understanding of the problem.Since we are looking for two numbers, we let

s = the number of student ticketsn = the number of non-student tickets

1. UNDERSTAND (continued)

continued

continued

2. TRANSLATE

Hilton University Drama club sold 311 tickets for a play.

s + n = 311

total receipts were $385.50

0.50s

Total receipts

= 385.50

Admission for students

1.50n

Admission for non-students

+

continued

3. SOLVE

continued

We are solving the system s + n = 3110.50s + 1.50n = 385.50

Since the equations are written in standard form (and we might like to get rid of the decimals anyway), we’ll solve by the addition method. Multiply the second equation by –2.

s + n = 311

–2(0.50s + 1.50n) = –2(385.50)s + n = 311

–s – 3n = –771

–2n = –460 n = 230

continued

4. INTERPRET

Check: Substitute s = 81 and n = 230 into both of the equations. s + n = 311 First Equation

81 + 230 = 311 True0.50s + 1.50n = 385.50 Second Equation

40.50 + 345 = 385.50 True

0.50(81) + 1.50(230) = 385.50

State: There were 81 student tickets and 230 non student tickets sold.

Now we substitute 230 for n into the first equation to solve for s. s + n = 311

s + 230 = 311

s = 81

continued

Example

continued

Terry Watkins can row about 10.6 kilometers in 1 hour downstream and 6.8 kilometers upstream in 1 hour. Find how fast he can row in still water, and find the speed of the current.

1. UNDERSTAND

Read and reread the problem. We are going to propose a solution, but first we need to understand the formulas we will be using. Although the basic formula is d = r • t (or r • t = d), we have the effect of the water current in this problem. The rate when traveling downstream would actually be r + w and the rate upstream would be r – w, where r is the speed of the rower in still water, and w is the speed of the water current.

continued

1. UNDERSTAND (continued)

Suppose Terry can row 9 km/hr in still water, and the water current is 2 km/hr. Since he rows for 1 hour in each direction, downstream would be (r + w)t = d or (9 + 2)1 = 11 km

Upstream would be (r – w)t = d or (9 – 2)1 = 7 km

Our proposed solution is incorrect (hey, we were pretty close for a guess out of the blue), but we now have a better understanding of the problem.

Since we are looking for two rates, we let

r = the rate of the rower in still water

w = the rate of the water current continued

2. TRANSLATE

continued

rate downstream

(r + w)

time downstream

• 1

distance downstream

= 10.6

rate upstream

(r – w)

time upstream

• 1

distance upstream

= 6.8continued

3. SOLVE

continued

continued

We are solving the system r + w = 10.6

r – w = 6.8

Since the equations are written in standard form, we’ll solve by the addition method. Simply add the two equations together. r + w = 10.6

r – w = 6.8 2r = 17.4

r = 8.7

4. INTERPRET

continued

Check: Substitute r = 8.7 and w = 1.9 into both equations.

(r + w)1 = 10.6 First equation (8.7 + 1.9)1 = 10.6 True

(r – w)1 = 1.9 Second equation

(8.7 – 1.9)1 = 6.8 True

State: Terry’s rate in still water is 8.7 km/hr and the rate of the water current is 1.9 km/hr.

Now we substitute 8.7 for r into the first equation.r + w = 10.6

8.7 + w = 10.6w = 1.9

Example

continued

A Candy Barrel shop manager mixes M&M’s worth $2.00 per pound with trail mix worth $1.50 per pound. How many pounds of each should she use to get 50 pounds of a party mix worth $1.80 per pound?

1. UNDERSTANDRead and reread the problem. We are going to propose a solution, but first we need to understand the formulas we will be using. To find out the cost of any quantity of items we use the formula

price per unit • number of units = price of all units

continued

continued

1. UNDERSTAND (continued)

Suppose the manage decides to mix 20 pounds of M&M’s. Since the total mixture will be 50 pounds, we need 50 – 20 = 30 pounds of the trail mix. Substituting each portion of the mix into the formula,M&M’s $2.00 per lb • 20 lbs = $40.00

trail mix $1.50 per lb • 30 lbs = $45.00

Mixture $1.80 per lb • 50 lbs = $90.00

continued

continued

1. UNDERSTAND (continued)

Since $40.00 + $45.00 ≠ $90.00, our proposed solution is incorrect (hey, we were pretty close again), but we now have a better understanding of the problem. Since we are looking for two quantities, we let

x = the amount of M&M’sy = the amount of trail mix

continued

2. TRANSLATE

continued

Fifty pounds of party mix

x + y = 50

price per unit • number of units = price of all unitsUsing

Price of M&M’s

2x

Price of trail mix

+ 1.5y

Price of mixture

= 1.8(50) = 90

3. SOLVE

continued

continued

We are solving the system x + y = 502x + 1.50y = 90

Since the equations are written in standard form, we’ll solve by the addition method. Multiply the first equation by 3 and the second equation by –2 (which will also get rid of the decimal).

3(x + y) = 3(50)

–2(2x + 1.50y) = –2(90)

3x + 3y = 150

–4x – 3y = –180–x = –30 x = 30

4. INTERPRET

continued

Check: Substitute x = 30 and y = 20 into both of the equations. x + y = 50 First equation

30 + 20 = 50 True

2x + 1.50y = 90 Second equation

60 + 30 = 90 True

2(30) + 1.50(20) = 90

State: The store manager needs to mix 30 pounds of M&M’s and 20 pounds of trail mix to get the mixture at $1.80 a pound.

Now we substitute 30 for x into the first equation.x + y = 50

30 + y = 50y = 20

Solving a Problem with 3 Variables

Continued

The measure of the largest angle of a triangle is 90o more than the measure of the smallest angle, and the measure of the remaining angle is 30o more than the measure of the smallest angle. Find the measure of each angle.

UnderstandRead and reread the problem. We are going to propose a solution.Suppose the measure of the smallest angle is 30º. Then the largest angle is 30º + 90º = 120º (90º more than the smallest), and the remaining angle is 30º + 30º = 60º (30º more than the smallest).In a triangle, the sum of the measures of the 3 angles is 180º. Since 30º + 120º + 60º 180º, our proposed solution is incorrect, but we now have a better understanding of the problem.

Continued

Translate

continued

If we let s = the measure of the smallest angle, then m = the measure of the middle angle, and l = the measure of the largest angle, then

we are solving the system s + m + l = 180

l = 90 + s m = 30 + s

Solve

Continued

continued

Substitute the last two equations into the first equation.

s + 30 + s + 90 + s = 180

3s + 120 = 180

3s = 60

s = 20

Substitute this value for s into the 2nd and 3rd equations from the original set.

l = 90 + s = 90 + 20 = 110

m = 30 + s = 30 + 20 = 50

Interpret

continued

Check: Substitute s = 20, m = 50, and l = 110 into all three equations.

20 + 50 + 110 = 180 true 110 = 90 + 20 true 50 = 30 + 20 true

State: The 3 angles of the triangle are 20º, 50º and 110º.