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LINEAR SYSTEMS – Word Problems
There are 3 types of problems we will look at :
1. Plane / Boat problems
2. Money problems
3. Number problems
LINEAR SYSTEMS – Word Problems
There are 3 types of problems we will look at :
1. Plane / Boat problems
2. Money problems
3. Number problems
PLANE / BOAT problems
ALWAYS use this format :
distance
x y 
short time
distance
x y 
long time
LINEAR SYSTEMS – Word Problems
There are 3 types of problems we will look at :
1. Plane / Boat problems
2. Money problems
3. Number problems
PLANE / BOAT problems
ALWAYS use this format
distance
x y 
short time
distance
x y 
long time
Once we fill in the given information, use the
addition method to solve for ( x , y )
LINEAR SYSTEMS – Word Problems
EXAMPLE : A plane travels 600 miles in 2 hours with the wind. It makes the return
trip in 3 hours against the wind. Find the speed of the plane “x “ in no
wind, and “y” the speed of the wind
PLANE / BOAT problems
ALWAYS use this format
distance
x y 
short time
distance
x y 
long time
Fill in the given information…
LINEAR SYSTEMS – Word Problems
EXAMPLE : A plane travels 600 miles in 2 hours with the wind. It makes the return
trip in 3 hours against the wind. Find the speed of the plane “x “ in no
wind, and “y” the speed of the wind
PLANE / BOAT problems
- distance = 600
600
x y 
short time
600
x y 
long time
Fill in the given information…
LINEAR SYSTEMS – Word Problems
EXAMPLE : A plane travels 600 miles in 2 hours with the wind. It makes the return
trip in 3 hours against the wind. Find the speed of the plane “x “ in no
wind, and “y” the speed of the wind
PLANE / BOAT problems
- short time = 2 hours
600
x y 
2
600
x y 
long time
Fill in the given information…
LINEAR SYSTEMS – Word Problems
EXAMPLE : A plane travels 600 miles in 2 hours with the wind. It makes the return
trip in 3 hours against the wind. Find the speed of the plane “x “ in no
wind, and “y” the speed of the wind
PLANE / BOAT problems
- long time = 3 hours
Fill in the given information…
600
x y 
2
600
x y 
3
LINEAR SYSTEMS – Word Problems
EXAMPLE : A plane travels 600 miles in 2 hours with the wind. It makes the return
trip in 3 hours against the wind. Find the speed of the plane “x “ in no
wind, and “y” the speed of the wind
PLANE / BOAT problems
300
600
x y 
2
600
x y 
3
200
Simplify the fractions…
LINEAR SYSTEMS – Word Problems
EXAMPLE : A plane travels 600 miles in 2 hours with the wind. It makes the return
trip in 3 hours against the wind. Find the speed of the plane “x “ in no
wind, and “y” the speed of the wind
PLANE / BOAT problems
x  y  300
x  y  200
Our new equations …
LINEAR SYSTEMS – Word Problems
EXAMPLE : A plane travels 600 miles in 2 hours with the wind. It makes the return
trip in 3 hours against the wind. Find the speed of the plane “x “ in no
wind, and “y” the speed of the wind
PLANE / BOAT problems
+
x  y  300
x  y  200
2x
 500
Addition method …
LINEAR SYSTEMS – Word Problems
EXAMPLE : A plane travels 600 miles in 2 hours with the wind. It makes the return
trip in 3 hours against the wind. Find the speed of the plane “x “ in no
wind, and “y” the speed of the wind
PLANE / BOAT problems
x  y  300
+
x  y  200
2x
2
Solve for “x” …
500

2
x  250
Speed of plane in no wind…
LINEAR SYSTEMS – Word Problems
EXAMPLE : A plane travels 600 miles in 2 hours with the wind. It makes the return
trip in 3 hours against the wind. Find the speed of the plane “x “ in no
wind, and “y” the speed of the wind
PLANE / BOAT problems
x  y  300
250  y  300
 250
 250
y  50
x  250
Speed of the wind
Speed of plane in no wind…
Substitute to find “y”
LINEAR SYSTEMS – Word Problems
There are 3 types of problems we will look at :
1. Plane / Boat problems
2. Money problems
3. Number problems
MONEY problems
Count Equation
how many items were sold ?
x  y  total count of items
Money Equation
what are the prices of the items AND
how much $$$ was collected ?
price x  price  y   total $$$ collected
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The boy scouts sold chips for $0.50 and pretzels for $0.75 at a recent
camping outing. If they collected $94.50 and sold 154 items, how
many bags of chips ( x ) and pretzels ( y ) were sold ?
MONEY problems
Count Equation
how many items were sold ?
x  y  total count of items
Money Equation
what are the prices of the items AND
how much $$$ was collected ?
price x  price  y   total $$$ collected
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The boy scouts sold chips for $0.50 and pretzels for $0.75 at a recent
camping outing. If they collected $94.50 and sold 154 items, how
many bags of chips ( x ) and pretzels ( y ) were sold ?
MONEY problems
Find the given information and place in the correct equations…
Count Equation
how many items were sold ?
x  y  total count of items
Money Equation
what are the prices of the items AND
how much $$$ was collected ?
price x  price  y   total $$$ collected
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The boy scouts sold chips for $0.50 and pretzels for $0.75 at a recent
camping outing. If they collected $94.50 and sold 154 items, how
many bags of chips ( x ) and pretzels ( y ) were sold ?
x  y  154
MONEY problems
Find the given information and place in the correct equations…
Count Equation
how many items were sold ?
x  y  total count of items
Money Equation
what are the prices of the items AND
how much $$$ was collected ?
price x  price  y   total $$$ collected
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The boy scouts sold chips for $0.50 and pretzels for $0.75 at a recent
camping outing. If they collected $94.50 and sold 154 items, how
many bags of chips ( x ) and pretzels ( y ) were sold ?
x  y  154
0.50 x  0.75 y  94.50
MONEY problems
Find the given information and place in the correct equations…
Count Equation
how many items were sold ?
x  y  total count of items
Money Equation
what are the prices of the items AND
how much $$$ was collected ?
price x  price  y   total $$$ collected
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The boy scouts sold chips for $0.50 and pretzels for $0.75 at a recent
camping outing. If they collected $94.50 and sold 154 items, how
many bags of chips ( x ) and pretzels ( y ) were sold ?
x  y  154
0.50 x  0.75 y  94.50
I like to use the
addition method
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The boy scouts sold chips for $0.50 and pretzels for $0.75 at a recent
camping outing. If they collected $94.50 and sold 154 items, how
many bags of chips ( x ) and pretzels ( y ) were sold ?
 0.50x  y  154
0.50 x  0.75 y  94.50
 0.50 x  0.50 y  77.00
Multiplied top
equation by – 0.50
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The boy scouts sold chips for $0.50 and pretzels for $0.75 at a recent
camping outing. If they collected $94.50 and sold 154 items, how
many bags of chips ( x ) and pretzels ( y ) were sold ?
 0.50 x  y  154 
+
0.50 x  0.75 y  94.50
 0.50 x  0.50 y  77.00
0.25 y  17.50
Add the two
equations
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The boy scouts sold chips for $0.50 and pretzels for $0.75 at a recent
camping outing. If they collected $94.50 and sold 154 items, how
many bags of chips ( x ) and pretzels ( y ) were sold ?
+
 0.50x  y  154
0.50 x  0.75 y  94.50
 0.50 x  0.50 y  77.00
0.25 y  17.50
0.25 y 17.50

0.25
0.25
Solve for “y”
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The boy scouts sold chips for $0.50 and pretzels for $0.75 at a recent
camping outing. If they collected $94.50 and sold 154 items, how
many bags of chips ( x ) and pretzels ( y ) were sold ?
+
 0.50 x  y  154 
0.50 x  0.75 y  94.50
 0.50 x  0.50 y  77.00
0.25 y  17.50
0.25 y 17.50

0.25
0.25
y  70
Solve for “y”
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The boy scouts sold chips for $0.50 and pretzels for $0.75 at a recent
camping outing. If they collected $94.50 and sold 154 items, how
many bags of chips ( x ) and pretzels ( y ) were sold ?
+
 0.50 x  y  154 
0.50 x  0.75 y  94.50
 0.50 x  0.50 y  77.00
0.25 y  17.50
0.25 y 17.50

0.25
0.25
y  70
x  y  154
x  70  154
x  84
Substitute for “y” into either
original equation and solve for
“x”…
LINEAR SYSTEMS – Word Problems
There are 3 types of problems we will look at :
1. Plane / Boat problems
2. Money problems
3. Number problems
When working on number problems look for key words such as sum, total,
difference, twice, even / odd , etc. to create your equations. Then use either the
substitution method or addition method to find your answers.
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The sum of two numbers is 24. Their difference is 16.
Find the two numbers
When working on number problems look for key words such as sum, total,
difference, twice, even / odd , etc. to create your equations. Then use either the
substitution method or addition method to find your answers.
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The sum of two numbers is 24. Their difference is 16.
Find the two numbers
x  y  24
When working on number problems look for key words such as sum, total,
difference, twice, even / odd , etc. to create your equations. Then use either the
substitution method or addition method to find your answers.
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The sum of two numbers is 24. Their difference is 16.
Find the two numbers
x  y  24
x  y  16
When working on number problems look for key words such as sum, total,
difference, twice, even / odd , etc. to create your equations. Then use either the
substitution method or addition method to find your answers.
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The sum of two numbers is 24. Their difference is 16.
Find the two numbers
+
x  y  24
x  y  16
2 x  40
Addition method…
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The sum of two numbers is 24. Their difference is 16.
Find the two numbers
+
x  y  24
x  y  16
2 x  40
Solve for “x”
2 x 40

2
2
x  20
LINEAR SYSTEMS – Word Problems
EXAMPLE :
The sum of two numbers is 24. Their difference is 16.
Find the two numbers
+
x  y  24
x  y  16
2 x  40
2 x 40

2
2
x  20
Substitute for “x” in
either equation and
solve for “y”…
x  y  24
20  y  24
y4
LINEAR SYSTEMS – Word Problems
EXAMPLE # 2 :
The sum of two numbers is 27. The second number is three more
than twice the first number.
LINEAR SYSTEMS – Word Problems
EXAMPLE # 2 :
The sum of two numbers is 27. The second number is three
more than twice the first number.
x  y  27
LINEAR SYSTEMS – Word Problems
EXAMPLE # 2 :
The sum of two numbers is 27. The second number is three
more than twice the first number.
x  y  27
y  2x  3
LINEAR SYSTEMS – Word Problems
EXAMPLE # 2 :
The sum of two numbers is 27. The second number is three
more than twice the first number.
x  y  27
y  2x  3
I will use substitution method…
x  2x  3  27
LINEAR SYSTEMS – Word Problems
EXAMPLE # 2 :
The sum of two numbers is 27. The second number is three
more than twice the first number.
x  y  27
y  2x  3
x  2x  3  27
x  2 x  3  27
3x  3  27
 3  3
+
3x  24
x 8
Solve for “x”…
LINEAR SYSTEMS – Word Problems
EXAMPLE # 2 :
The sum of two numbers is 27. The second number is three
more than twice the first number.
x  y  27
y  2x  3
x  2x  3  27
x  2 x  3  27
3x  3  27
 3  3
+
3x  24
x 8
Substitute for
“x” and solve
for “y’…
8  y  27
y  19
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