Modelling with systems of equations Problem set B Worked solutions

Three students buy supplies at a school shop. Zoe buys 4 pens, 3 notebooks and 2 rulers and pays $13.90. Yusuf buys 2 pens, 5 notebooks and 1 ruler and pays $15.35. Xan buys 3 pens, 1 notebook and 6 rulers and pays $11.25. Find the cost of one ruler.

• Let \(x\) = cost of one pen ($), \(y\) = cost of one notebook ($), \(z\) = cost of one ruler ($).
• \(4x + 3y + 2z=13.9\)
• \(2x + 5y + z=15.35\)
• \(3x + y + 6z=11.25\)
• Simultaneous equation solver: x = 1.25, y = 2.40, z = 0.850
• One ruler costs $0.85.

Solve the system and find all possible values of \(x\), \(y\) and \(z\): \(x^2 + 2y^2 + z^2=42\), \(3x^2 - y^2 + z^2=80\), \(x^2 + y^2 - 2z^2=11\).

• Let \(p=x^2\), \(q=y^2\), \(r=z^2\). The system is linear in \(p, q, r\).
• \(p + 2q + r=42\)
• \(3p - q + r=80\)
• \(p + q - 2r=11\)
• Simultaneous equation solver: p = 25, q = 4, r = 9
• Take square roots, positive and negative: x = ±5, y = ±2, z = ±3

A chemist mixes three salt solutions to make 600 ml of a mixture. Solution X is 10% salt, solution Y is 25% salt and solution Z is 40% salt. There is 100 ml more of solution Y than of solution X. The mixture contains 180 ml of salt. How much of solution Z is used?

• Let \(x\) = volume of solution X (ml), \(y\) = volume of solution Y (ml), \(z\) = volume of solution Z (ml).
• \(x + y + z=600\)
• \(0.1x + 0.25y + 0.4z=180\)
• \(-x + y=100\)
• Simultaneous equation solver: x = 100, y = 200, z = 300
• The amount of solution Z is 300 ml.

A juice bar makes three kinds of smoothie. A classic uses 120 g of mango and 150 g of yoghurt. A tropical uses 80 g of mango and 60 g of yoghurt. A green uses 60 g of mango and 40 g of yoghurt. The classics sell for $4.50 each, the tropicals for $3.80 each and the greens for $5.20 each. In one day all the ingredients used are 6.6 kg of mango and 6.66 kg of yoghurt, and the total sales are $319.80. How many tropicals are made?

• Let \(x\) = number of classics, \(y\) = number of tropicals, \(z\) = number of greens.
• Work in grams: 6.6 kg = 6 600 g and 6.66 kg = 6 660 g.
• \(120x + 80y + 60z=6600\)
• \(150x + 60y + 40z=6660\)
• \(4.5x + 3.8y + 5.2z=319.8\)
• Simultaneous equation solver: x = 30, y = 24, z = 18
• 24 tropicals are made.

One watermelon has the same mass as 5 pineapples, and the same mass as 20 kiwi fruit. 3 watermelons and 4 pineapples have a mass 5 520 g more than 30 kiwi fruit. Find the total mass of one watermelon, one pineapple and one kiwi fruit.

• Let \(p\) = mass of one watermelon (g), \(q\) = mass of one pineapple (g), \(r\) = mass of one kiwi fruit (g).
• \(p-5q=0\) (one watermelon is 5 pineapples)
• \(p-20r=0\) (one watermelon is 20 kiwi fruit)
• \(3p + 4q - 30r=5520\)
• Simultaneous equation solver: p = 2400, q = 480, r = 120
• Total = 2 400 + 480 + 120 = 3 000 g

A three-digit number has digits \(a\), \(b\) and \(c\), in that order, so the number is \(100a+10b+c\). The digits add up to 17. The middle digit is 2 more than the last digit. When the digits are reversed, the number decreases by 594. Find the number.

• Let \(a\) = the first digit, \(b\) = the middle digit, \(c\) = the last digit.
• \(a + b + c=17\)
• \(b - c=2\)
• \(99c-99a=-594\) (the reversed number minus the number is −594)
• Simultaneous equation solver: a = 9, b = 5, c = 3
• The digits are 9, 5, 3: the number is 953.

Three tables order breakfast at a bakery. Table 1 buys 3 croissants, 2 buns and 2 pastries and pays $15.50. Table 2 buys 1 croissant, 4 buns and 3 pastries and pays $16.90. Table 3 buys 5 croissants, 1 bun and 1 pastry and pays $15.45. Find the total cost of one croissant, one bun and one pastry.

• Let \(x\) = cost of one croissant ($), \(y\) = cost of one bun ($), \(z\) = cost of one pastry ($).
• \(3x + 2y + 2z=15.5\)
• \(x + 4y + 3z=16.9\)
• \(5x + y + z=15.45\)
• Simultaneous equation solver: x = 2.20, y = 1.35, z = 3.10
• x + y + z = 2.20 + 1.35 + 3.10 = $6.65

A school club survey records the activity chosen by students. The pie chart shows how students chose between music, drama and chess. The sector for music has angle 90°. There are 4 more in drama than in chess. The number in music and chess together is 6 more than the number in drama. One person is chosen at random. Find the probability that they chose drama.

• Let \(x\), \(y\), \(z\) be the numbers who chose music, drama and chess.
• \(y-z=4\)
• \(x-y+z=6\)
• The sector for music is 90/360 of the whole: \(\frac{x}{x+y+z}=\frac{90}{360}\), so \(3x - y - z=0\)
• Simultaneous equation solver: x = 10, y = 17, z = 13
• Total = 40. Probability = \(\frac{17}{40}\) = 0.425

Three numbers \(a\), \(b\), \(c\) are in order, \(a\le b\le c\). The mean equals the median. The range is 3 more than the median. The difference between the two largest numbers is 5. Find the largest of the three numbers.

• The median is \(b\), the mean is \(\frac{a+b+c}{3}\) and the range is \(c-a\).
• \(a - 2b + c=0\)
• \(-a - b + c=3\)
• \(-b + c=5\)
• Simultaneous equation solver: a = 2, b = 7, c = 12
• The answer is 12.