AI SL Ā· 12.4
0 of 45 skills done
Modelling with systems of equations
12.4.2
Screenshot and video to come
Screenshot and video to come
Three friends buy lunch at a canteen. Priya buys 3 sandwiches, 2 soups and 4 juices and pays $20.00. Quinn buys 2 sandwiches, 5 soups and 1 juice and pays $20.95. Ravi buys 4 sandwiches and 3 juices and pays $17.05. Find the cost of one soup.
Have a go first. Then open the working one line at a time.
Let x = sandwich, y = soup, z = juice (all in $)
Say what each letter stands for.
3x + 2y + 4z = 20.00; 2x + 5y + z = 20.95; 4x + 0y + 3z = 17.05
Ravi buys no soup, so his y-coefficient is 0.
Simultaneous equation solver: x = 3.40, y = 2.60, z = 1.15
Enter the three rows of coefficients and the three totals.
One soup costs $2.60
The solver gives all three prices. Give the one that was asked for.
Meera invests $12 000 in three accounts for one year. Account A pays 1.5% per year, account B pays 2.5% per year and account C pays 4% per year. There is $1 000 more in account B than in account A. The total interest earned over the year is $345.00. How much is in account C?
Have a go first. Then open the working one line at a time.
Let a, b, c = amounts in accounts A, B, C ($)
Each letter is an amount of money.
a + b + c = 12 000
The three amounts add to the whole sum.
0.015a + 0.025b + 0.04c = 345
Change each rate from a percentage to a decimal: 1.5% = 0.015. One year, so the interest is rate Ć amount.
āa + b = 1000
B is $1000 more than A, so b ā a = 1000.
Simultaneous equation solver: a = 3000, b = 4000, c = 5000
Give the amount in C: $5000.
A bakery makes muffins, scones and tarts. A muffin uses 120 g of flour and 20 g of butter. A scone uses 180 g of flour and 40 g of butter. A tart uses 150 g of flour and 90 g of butter. The muffins sell for $2.20 each, the scones for $1.80 each and the tarts for $3.50 each. In one day all the ingredients used are 13.2 kg of flour and 5 kg of butter, and the total sales are $242.00. How many tarts are made?
Have a go first. Then open the working one line at a time.
Let x, y, z = number of muffins, scones, tarts
The unknowns are counts, not prices.
120x + 180y + 150z = 13 200
13.2 kg = 13 200 g. Use the same unit on both sides.
20x + 40y + 90z = 5000
5 kg = 5000 g.
2.2x + 1.8y + 3.5z = 242
Price Ć number for each item, added.
Simultaneous equation solver: x = 30, y = 20, z = 40
There are 40 tarts.
At a shop one sofa costs the same as 5 chairs, and the same as 12 stools. 4 sofas and 9 chairs cost $248 more than 20 stools. Find the total cost of one sofa, one chair and one stool.
Have a go first. Then open the working one line at a time.
Let p = sofa, q = chair, r = stool (in $)
Three prices, so three equations.
p = 5q ā p ā 5q = 0; p = 12r ā p ā 12r = 0
"Costs the same as" gives an equation. Move everything to the left.
4p + 9q ā 20r = 248
"$248 more than" means (cost of the sofas and chairs) ā (cost of the stools) = 248.
Simultaneous equation solver: p = 60, q = 12, r = 5
Then add: 60 + 12 + 5 = 77.
A sports club asks its members to choose a sport. The pie chart shows how members chose between football, swimming and tennis. The sector for football has angle 160Ā°. There are 5 more in swimming than in tennis. The number in football and tennis together is 7 more than the number in swimming. One person is chosen at random. Find the probability that they chose tennis.
Have a go first. Then open the working one line at a time.
Let x, y, z = number choosing football, swimming, tennis
Use counts, then turn them into a probability at the end.
y ā z = 5; x + z ā y = 7 ā x ā y + z = 7
Each sentence about the numbers gives an equation.
x / (x + y + z) = 160/360 ā 200x ā 160y ā 160z = 0 ā 5x ā 4y ā 4z = 0
A 160Ā° sector is 160/360 of the whole. Multiply out and divide by 40.
Simultaneous equation solver: x = 12, y = 10, z = 5
The total is 12 + 10 + 5 = 27.
P(tennis) = 5/27 = 0.185
Favourite count over the total.
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 15. The middle digit is 1 less than twice the first digit. When the digits are reversed, the number increases by 396. Find the number.
Have a go first. Then open the working one line at a time.
The digits are a, b, c, so the number is 100a + 10b + c
The place value matters. The reversed number is 100c + 10b + a.
a + b + c = 15; b = 2a ā 1 ā 2a ā b = 1
Write each condition as an equation.
(100c + 10b + a) ā (100a + 10b + c) = 396 ā 99c ā 99a = 396
The 10b terms cancel.
Simultaneous equation solver: a = 3, b = 5, c = 7
Write the digits as the number: 357.
Solve the system and find all possible values of \(x\), \(y\) and \(z\): \(x^2 + y^2 + z^2=38\), \(x^2 + 3y^2 + 2z^2=71\), \(2x^2 - y^2 + z^2=39\).
Have a go first. Then open the working one line at a time.
Let p = xĀ², q = yĀ², r = zĀ²
The system is linear in p, q and r.
p + q + r = 38; p + 3q + 2r = 71; 2p ā q + r = 39
Same coefficients, new letters.
Simultaneous equation solver: p = 9, q = 4, r = 25
These are xĀ², yĀ² and zĀ².
x = Ā±3, y = Ā±2, z = Ā±5
Each square root has a positive and a negative value.
Three numbers \(a\), \(b\), \(c\) are in order, \(a\le b\le c\). The mean is 1 more than the median. The range is 4 more than the median. The difference between the largest number and the median is equal to the median. Find the largest of the three numbers.
Have a go first. Then open the working one line at a time.
median = b; mean = (a + b + c)/3; range = c ā a
Because a ā¤ b ā¤ c, the median is the middle number b.
(a + b + c)/3 = b + 1 ā a ā 2b + c = 3
Multiply by 3, then move 3b across.
c ā a = b + 4 ā āa ā b + c = 4; c ā b = b ā ā2b + c = 0
The third equation has no a: its coefficient is 0.
Simultaneous equation solver: a = 3, b = 7, c = 14
The largest number is 14.