Systems of 3 Variables

1

100

Solve the simultaneous equations

5x + 3y = 41 

2x + 3y = 20  

x = 7

y = 2

200

David buys 2 scones and 2 coffees in a shop and the cost is £18. Ellie buys 3 scones and 2 coffees in the same shop and they cost £22. Form two equations and solve to find the cost of each scone and each coffee.

s = £4

c = £5

300

Solve the simultaneous equations

x − y + 3z = 5 

x + y + 6z = 12 

3x − 2y + 2z = 10

x = 4

y = 2 

z = 1

400

The community college theater department sold three kinds of tickets to its latest play production.

• The adult tickets sold for $15, the student tickets for $10 and the child tickets for $8. 
• The theater department was thrilled to have sold 250 tickets and brought in $2,825 in one night. 
• The number of student tickets sold is twice the number of adult tickets sold. 

How many of each type did the department sell?

The theater department sold 75 adult tickets,
150 student tickets, and 25 child tickets.

500

Marina had $24,500 to invest. She divided the money into three different accounts. 

At the end of the year, she had made $1,300 in interest. 

The annual yield on each of the three accounts was 4% (y1), 5.5% (y2), and 6% (y3).

If the amount of money in the 4% account was four times the amount of money in the 5.5% account, how much had she placed in each account?

4% account (y1): $8,000

5.5% account (y2): $2,000

6% account (y3): $14,500

2

100

Solve the simultaneous equations

2x + 4y = 26 

3x − y = 4

x = 3

y = 5

200

Alan and Connor have £6.70 in total. Alan has £1.70 more than Connor. Let a be the amount of money Alan has. Let c be the amount of money Connor has. Set up a pair of simultaneous equations and solve to find out how much each person has.

a = £4.20

c = £2.50

300

Solve the simultaneous equations

2x + 3y + 5z = 21 

3x + 6y + 15z = 51 

5x + 4y + 10z = 37

x = 1

y = 3 

z = 2

400

A local bookstore tracks its inventory of three popular genres: Fiction (F), Mystery (M), and Science Fiction (SF). At the end of the month, the following information is available:

• The bookstore has a total of 350 books from these three genres in stock: F + M + SF = 350.
• Based on their average prices and total value, the combined 'value points' of these books is 4650. Fiction books have a value point of 20, Mystery books have a value point of 12, and Science Fiction books have a value point of 10. So, 20F + 12M + 10SF = 4650.
• The number of Fiction books in stock is the same as the number of Science Fiction books in stock: F = SF.

How many books of each genre (Fiction, Mystery, and Science Fiction) does the bookstore have in stock?

The bookstore sold 75 fiction books, 200 mystery books, and 75 science fiction books.

500

The currents running through an electrical system are given by the following system of equations.  The three currents, I1, I2, and I3, are measured in amps. Solve the system to find the currents in this circuit.

I1 + 2I2 - I3 = 0.425
3I1 - I2 + 2I3 = 2.225
5I1 + I2 + 2I3 = 3.775

I1 = 0.375 amps

I2 = 0.400 amps

I3 = 0.750 amps

3

100

Solve the simultaneous equations

x + 7y = 64 

x + 3y = 28

x = 1

y = 9

200

A museum sells adult tickets or child tickets. Fozia buys 4 adult tickets and 1 child ticket for £120 Sami buys 5 adult tickets and 3 child tickets for £171 Work out the cost of each type of ticket

a = £27

c = £12

300

Solve the simultaneous equations 

2x + 4y − z = 15 

3x + 8y + z = 44 

x + 2y + 2z = 15

x = -5

y = 7

z = 3

400

A local market sells three types of fruit: Apples (A), Bananas (B), and Cherries (C). Here's what we know about a particular day's sales:

• A total of 600 pieces of fruit were sold
• Apples sell for $10, Bananas sell for $8 and Cherries sell for $5. The total revenue from the fruit sales was $4900
• The number of Apples sold is twice the number of Cherries sold

How many of each type of fruit were sold at the market?

The market sold 200 Apples, 300 Bananas, and 100 Cherries.

500

Find the equation of the parabola y = ax² + bx + c that passes through the following three points:
(-2, 40), (1, 7), (3, 15).

a = 3

b = −8

c = 12

4

100

Solve the simultaneous equations

4x − 4y = 24 

x − 4y = 3

x = 7

y = 1

200

Three bananas and two pears cost £2.07 Five bananas and three pears cost £3.33 Find the cost of ten bananas and ten pears.

£8.10

300

Solve the simultaneous equations

10x + 60y + 10z = 25 

5x + 40y + 20z = 40 

20x + 20y + 40z = 30

x = -2

y = 0.5

z = 1.5

400

A farmer is preparing a special feed mix for their animals. The mix contains three types of grain: Oats (O), Corn (C), and Barley (B).

• The total weight of the feed mix is 500 kilograms
• The cost per kilogram of each grain is: Oats - $0.50/kg, Corn - $0.80/kg, Barley - $0.60/kg. The total cost of the 500 kg mix is $330
• The amount of Corn in the mix is three times the amount of Oats

How many kilograms of each type of grain are in the feed mix?

The feed mix contains 60 kilograms of Oats, 180 kilograms of Corn, and 260 kilograms of Barley.

500

In the position function for vertical height, s(t) = ½at2 + v0t + s0, s(t) represents height in meters and t represents time in seconds. 

A volleyball served at an initial height of one meter has a height of 6.275 meters ½ second after the serve and a height of 9.1 meters one second after the serve.

Find the values for a, v and s.

a = −9.8

v = 13

s = 1

5

100

Solve the simultaneous equations

2x + 4y = 14 

4x − 4y = 4

x = 3

y = 2

200

Albie is training for a marathon. He jogs either route A or route B. During April, he jogs route A nine times and route B five times. Route B is 8 miles longer than route A. In total, he jogs 89 miles in April. In May, he will start jogging route C. Route C is 20% longer than route B. Work out the length of route C.

Route C = 13.8 miles

300

Solve the simultaneous equations

x + y + z = 1 

4x − 3y + 4z = 32 

x − 10y − 2z = 27

x = -1

y = -4

z = 6

400

A jewelry store sells three types of earrings: Silver (S), Gold-plated (GP), and Gemstone (G).

• The store sold a total of 120 pairs of earrings during a special sale
• The selling price for each type of earring is: Silver - $25/pair, Gold-plated - $40/pair, Gemstone - $30/pair. The total revenue from the earring sales was $3950
• The number of Silver earrings sold was 20 more than the number of Gemstone earrings sold

How many pairs of each type of earring did the store sell?

The store sold 42 pairs of Silver earrings, 56 pairs of Gold-plated earrings, and 22 pairs of Gemstone earrings.

500

Consider a six-sided die with probabilities P(1), P(2), and P(3) for rolling a 1, 2, and 3, respectively. Due to the bias, we have the following information:

• The sum of the probabilities of rolling a 1, 2, or 3 is 0.6

• The probability of rolling a 2 is 0.1 more than the probability of rolling a 1

• The probability of rolling a 3 is equal to the sum of the probabilities of rolling a 1 and the probability of rolling a 2

Find the probability of rolling a 1, a 2, and a 3 for this biased die.

P(1) = 0.1

P(2) = 0.2

P(3) = 0.3