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.

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: Give a value of the constant c for which the system is inconsistent. 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 Protein A B 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 20 2 4 B 5 1 C

Each unit of food C has 80 g of carbohydrate, 3 g of fat and 8 g of protein 20 2 4 B 5 1 C 80 3 8

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

Write the equations Carbohydrate Fat Protein A 20 2 4 B 5 1 C 80 3 8 Total 140 11 24

Consider the following system of three equations in x, y and z. Example 5 Consider the following system of three equations in x, y and z. 4x + 3y + 2z = 11 3x + 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.

Add the first two equations and put it with the third equation Inconsistent Add the first two equations and put it with the third equation 4x + 3y + 2z = 11 3x + 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 = 11 3x + 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 = B Give 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 ≠ 14 2. consistent but with an infinite number of solutions. A = 3, B = 14