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Sample Questions 91587

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**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 $ How many of each type of flower was ordered?

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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 $ How many of each type of flower was ordered?

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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 r + 2.6d = They ordered mostly carnations, and 20 fewer roses than daisies. d – r = 20 The total order came to $ How many of each type of flower was ordered?

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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 r + 2.6d = They ordered mostly carnations, and 20 fewer roses than daisies. d – r = 20 The total order came to $ How many of each type of flower was ordered?

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**Solve using your calculator and answer in context**

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

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Example 2 If possible, solve the following system of equations and explain the geometrical significance of your answer.

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**Calculator will not give you an answer.**

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

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

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**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

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Example 2 Solve the system of equations using Gauss-Jordan Method

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Example Solve the system of equations using Gauss-Jordan Method

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Example Solve the system of equations using Gauss-Jordan Method

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Example Solve the system of equations using Gauss-Jordan Method

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**Example Solve the system of equations using Gauss-Jordan Method**

No solution

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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.

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**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

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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.

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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.

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**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.

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Carbohydrate Fat Protein A B C

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

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**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

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**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

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**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

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

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**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.

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**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 = x + 5y + 3z = 19 7x + 5y + az = b a = 3, b ≠19

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**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 = x + 5y + 3z = 19 7x + 5y + az = b a = 3, b = 19

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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.

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**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

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Equations in Two variables

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