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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?
c + r + d = 200
1.5c + 5.75r + 2.6d = 589.50
0c - r + d = 20
Solve using your calculator and
answer in context
• There were 80 carnations, 50 roses and 70
daisies ordered.
c + r + d = 200
1.5c + 5.75r + 2.6d = 589.50
0c - r + d = 20
Example 2
• If possible, solve the following system of
equations and explain the geometrical
significance of your answer.
x - 2y + z = 2
-2x + 3y + z = -4
2x - y - 7z = 2
Calculator will not give you an answer.
• If possible, solve the following system of
equations and explain the geometrical
significance of your answer.
x - 2y + z = 2
-2x + 3y + z = -4
2x - y - 7z = 2
Objective - To solve systems of linear equations in
three variables.
Solve.
x - 2y + z = 2
-2x + 3y + z = -4
2x - 4y + 2z = 4
+
-2x + 3y + z = -4
-y + 3z = 0
2x - y - 7z = 2
-2x + 3y + z = -4
+
2x - y - 7z = 2
2y -6z = -2
-y + 3z = 0
+
y - 3z = -1
0 = -1
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 GaussJordan Method
é
ê 1 -2 1
-2x + 3y + z = -4 ê -2 3 1
2x - y - 7z = 2 êë 2 -1 -7
x -2y+ z = 2
ù
2 ú
-4 ú
2 úû
2R1 + R2 ® R2
2 -4 2 4
-2 3 1 -4
0 -1 3 0
Example
• Solve the system of equations using GaussJordan Method
é
ê 1 -2 1
-2x + 3y + z = -4 ê 0 -1 3
2x - y - 7z = 2 êë 2 -1 -7
x -2y+ z = 2
ù
2 ú
0 ú
2 úû
-2 R1 + R3 ® R3
-2
4
-2
-4
2
-1
-7
2
0
3
-9
-2
Example
• Solve the system of equations using GaussJordan Method
é
ê 1 -2 1
-2x + 3y + z = -4 ê 0 -1 3
2x - y - 7z = 2 êë 0 3 -9
x -2y+ z = 2
ù
2 ú
0 ú
-2 úû
-R2 ® R2
0
1
-3
0
Example
• Solve the system of equations using GaussJordan Method
é
ê 1 -2 1
-2x + 3y + z = -4 ê 0 1 -3
2x - y - 7z = 2 êë 0 3 -9
x -2y+ z = 2
ù
2 ú
0 ú
-2 úû
-3R2 + R3 ® R3
0
0
0
-3
3
0
9
-9
0
0
-2
-2
Example
• Solve the system of equations using GaussJordan Method
é
ê 1 -2 1
-2x + 3y + z = -4 ê 0 1 -3
2x - y - 7z = 2 êë 0 0 0
x -2y+ z = 2
ù
2 ú
0 ú
-2 úû
No solution
Example 3
Consider the following system of two linear
equations, where c is a constant: 2x + 5y = 16
4x + cy = 25
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
2x + 5y = 16 Þ 4x +10y = 32
4x + cy = 25 Þ 4x + cy = 25
c = 10
If c is chosen so that the system is consistent, explain in
geometrical terms why there is a unique solution.
2x + 5y = 16
4x + cy = 25
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
A
B
C
Fat
Protein
Each unit of food A has 20 g of carbohydrate, 2 g of fat
and 4 g of protein
A
B
C
Carbohydrate
20
Fat
2
Protein
4
Each unit of food B has 5 g of carbohydrate, 1 g of fat
and 2 g of protein
A
B
C
Carbohydrate
20
5
Fat
2
1
Protein
4
2
Each unit of food C has 80 g of carbohydrate, 3 g of fat
and 8 g of protein
A
B
C
Carbohydrate
20
5
80
Fat
2
1
3
Protein
4
2
8
The dietician designs the special meal so that it contains 140 g
of carbohydrate, 11 g of fat and 24 g of protein.
A
B
C
Total
Carbohydrate
20
5
80
140
Fat
2
1
3
11
Protein
4
2
8
24
Write the equations
A
B
C
Total
Carbohydrate
20
5
80
140
Fat
2
1
3
11
20a + 5b + 80c = 140
2a + b + 3c = 11
4a + 2b + 8c = 24
Protein
4
2
8
24
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
• Ax + 5y + 6z = B
3x + 5y + 6z = 14
Ax + 5y + 6z = B
1. inconsistent. A = 3, B ≠ 14
2. consistent but with an infinite number of
solutions. A = 3, B = 14