Sample Questions

91587

Example 1

• Billy’s Restaurant ordered 200 flowers for Mother’s Day.

• They ordered carnations at $1.50 each, roses at $5.75 each, and daisies at $2.60 each.

• They ordered mostly carnations, and 20 fewer roses than daisies. The total order came to $589.50. How many of each type of flower was ordered?

Decide your variables

• Billy’s Restaurant ordered 200 flowers for Mother’s Day.

• They ordered carnations at $1.50 each, roses at $5.75 each, and daisies at $2.60 each.

• They ordered mostly carnations, and 20 fewer roses than daisies. The total order came to $589.50. How many of each type of flower was ordered?

Write the equations

• Billy’s Restaurant ordered 200 flowers for Mother’s Day. c + r + d = 200

• They ordered carnations at $1.50 each, roses at $5.75 each, and daisies at $2.60 each.

• 1.5c + 5.75r + 2.6d = 589.50 • They ordered mostly carnations, and 20 fewer

roses than daisies. d – r = 20• The total order came to $589.50. • How many of each type of flower was ordered?

Order the equations

• Billy’s Restaurant ordered 200 flowers for Mother’s Day. c + r + d = 200

• They ordered carnations at $1.50 each, roses at $5.75 each, and daisies at $2.60 each.

• 1.5c + 5.75r + 2.6d = 589.50 • They ordered mostly carnations, and 20 fewer

roses than daisies. d – r = 20• The total order came to $589.50. • How many of each type of flower was ordered?

Solve using your calculator and answer in context

• There were 80 carnations, 50 roses and 70 daisies ordered.

Example 2

• If possible, solve the following system of equations and explain the geometrical significance of your answer.

Calculator will not give you an answer.

• If possible, solve the following system of equations and explain the geometrical significance of your answer.

Objective - To solve systems of linear equations in three variables.

Solve.

There is no solution. The three planes form a tent shape and the lines of intersection of pairs of planes are parallel to one another

Inconsistent, No Solution

Example 2

• Solve the system of equations using Gauss-Jordan Method

Example

• Solve the system of equations using Gauss-Jordan Method

Example

• Solve the system of equations using Gauss-Jordan Method

Example

• Solve the system of equations using Gauss-Jordan Method

Example

• Solve the system of equations using Gauss-Jordan Method

No solution

Example 3

Consider the following system of two linear equations, where c is a constant: 1. Give a value of the constant c for which the

system is inconsistent. 2. If c is chosen so that the system is consistent,

explain in geometrical terms why there is a unique solution.

Give a value of the constant c for which the system is inconsistent.

The lines must be parallel but not a multiple of each other

c = 10

If c is chosen so that the system is consistent, explain in geometrical terms why there is a unique solution.

It means that the 2 lines must have different gradients so they intersect to give a unique solution.

Example 4• The Health Club serves a special meal consisting of three

kinds of food, A, B and C. Each unit of food A has 20 g of carbohydrate, 2 g of fat and 4 g of protein. Each unit of food B has 5 g of carbohydrate, 1 g of fat and 2 g of protein. Each unit of food C has 80 g of carbohydrate, 3 g of fat and 8 g of protein. The dietician designs the special meal so that it contains 140 g of carbohydrate, 11 g of fat and 24 g of protein. Let a, b and c be the number of units of food A, B and C f respectively) used in the special meal. Set up a system of 3 simultaneous equations relating a, b and c.

• Do not solve the equations.

For this type of problem it is easier if you make a table

• The Health Club serves a special meal consisting of three kinds of food, A, B and C. Each unit of food A has 20 g of carbohydrate, 2 g of fat and 4 g of protein. Each unit of food B has 5 g of carbohydrate, 1 g of fat and 2 g of protein. Each unit of food C has 80 g of carbohydrate, 3 g of fat and 8 g of protein. The dietician designs the special meal so that it contains 140 g of carbohydrate, 11 g of fat and 24 g of protein. Let a, b and c be the number of units of food A, B and C f respectively) used in the special meal. Set up a system of 3 simultaneous equations relating a, b and c.

• Do not solve the equations.

Carbohydrate Fat ProteinABC

Each unit of food A has 20 g of carbohydrate, 2 g of fat and 4 g of protein

Carbohydrate Fat ProteinA 20 2 4BC

Each unit of food B has 5 g of carbohydrate, 1 g of fat and 2 g of protein

Carbohydrate Fat ProteinA 20 2 4B 5 1 2C

Each unit of food C has 80 g of carbohydrate, 3 g of fat and 8 g of protein

Carbohydrate Fat ProteinA 20 2 4B 5 1 2C 80 3 8

The dietician designs the special meal so that it contains 140 g of carbohydrate, 11 g of fat and 24 g of protein.

Carbohydrate Fat ProteinA 20 2 4B 5 1 2C 80 3 8Total 140 11 24

Write the equations

Carbohydrate Fat ProteinA 20 2 4B 5 1 2C 80 3 8Total 140 11 24

Example 5

Consider the following system of three equations in x, y and z.

4x + 3y + 2z = 113x + 2y + z = 8 7x + 5y + az = b

Give values for a and b in the third equation which make this system:1. inconsistent, 2. consistent, but with an infinite number of solutions.

Inconsistent

Add the first two equations and put it with the third equation

4x + 3y + 2z = 113x + 2y + z = 8 7x + 5y + 3z = 19

7x + 5y + az = b

• a = 3, b ≠19

Consistent with an infinite number of solutions

Add the first two equations and put it with the third equation

4x + 3y + 2z = 113x + 2y + z = 8 7x + 5y + 3z = 19

7x + 5y + az = b

• a = 3, b = 19

Example 6

Consider the following system of three equations in x, y and z.• 2x + 2y + 2z = 9• x + 3y + 4z = 5 • Ax + 5y + 6z = BGive possible values of A and B in the third equation which make this system:1. inconsistent.2. consistent but with an infinite number of solutions.

Example 6

• 2x + 2y + 2z = 9 • x + 3y + 4z = 5 3x + 5y + 6z = 14• Ax + 5y + 6z = B Ax + 5y + 6z = B

1. inconsistent. A = 3, B ≠ 142. consistent but with an infinite number of solutions. A = 3, B = 14