Sample Questions

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 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.

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.

Write the equations

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.

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

Sample Questions