linear systems – word problems there are 3 types of problems we will look at : 1. plane / boat...
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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 :
timelong
distanceshort time
distance
yx
yx
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
timelong
distanceshort time
distance
yx
yx
Once we fill in the given information, use the addition method to solve for ( x , y )
LINEAR SYSTEMS – Word Problems
PLANE / BOAT problems
ALWAYS use this format
timelong
distanceshort time
distance
yx
yx
Fill in the given information…
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
LINEAR SYSTEMS – Word Problems
PLANE / BOAT problems
- distance = 600
timelong
600short time
600
yx
yx
Fill in the given information…
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
LINEAR SYSTEMS – Word Problems
PLANE / BOAT problems
- short time = 2 hours
timelong
6002
600
yx
yx
Fill in the given information…
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
LINEAR SYSTEMS – Word Problems
PLANE / BOAT problems
- long time = 3 hours
3
6002
600
yx
yx
Fill in the given information…
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
LINEAR SYSTEMS – Word Problems
PLANE / BOAT problems
3
6002
600
yx
yxSimplify the fractions…
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
300
200
LINEAR SYSTEMS – Word Problems
PLANE / BOAT problems
200
300
yx
yxOur new equations …
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
LINEAR SYSTEMS – Word Problems
PLANE / BOAT problems
500 2
200
300
x
yx
yxAddition method …
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
+
LINEAR SYSTEMS – Word Problems
PLANE / BOAT problems
250
2
500
2
2
200
300
x
x
yx
yxSolve for “x” …
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
+
Speed of plane in no wind…
LINEAR SYSTEMS – Word Problems
PLANE / BOAT problems
250
50
250 250
300250
300
x
y
y
yx
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
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 ?
Money Equation what are the prices of the items AND
how much $$$ was collected ?
items ofcount total yx
collected $$$ totalpriceprice yx
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 ?
Money Equation what are the prices of the items AND
how much $$$ was collected ?
items ofcount total yx
collected $$$ totalpriceprice yx
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 ?
Money Equation what are the prices of the items AND
how much $$$ was collected ?
items ofcount total yx
collected $$$ totalpriceprice yx
Find the given information and place in the correct 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 ?
MONEY problems
Count Equation how many items were sold ?
Money Equation what are the prices of the items AND
how much $$$ was collected ?
items ofcount total yx
collected $$$ totalpriceprice yx
Find the given information and place in the correct equations…
154 yx
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 ?
Money Equation what are the prices of the items AND
how much $$$ was collected ?
items ofcount total yx
collected $$$ totalpriceprice yx
Find the given information and place in the correct equations…
94.500.750.50
154
yx
yx
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 ?
I like to use the addition method94.500.750.50
154
yx
yx
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 ?
00.770.500.50
94.500.750.50
154 0.50
yx
yx
yxMultiplied 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 ?
50.1725.0
00.770.500.50
94.500.750.50
154 0.50
y
yx
yx
yx
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 ?
25.0
50.17
25.0
25.0
50.1725.0
00.770.500.50
94.500.750.50
154 0.50
y
y
yx
yx
yx
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 ?
70
25.0
50.17
25.0
25.0
50.1725.0
00.770.500.50
94.500.750.50
154 0.50
y
y
y
yx
yx
yx
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 ?
70
25.0
50.17
25.0
25.0
50.1725.0
00.770.500.50
94.500.750.50
154 0.50
y
y
y
yx
yx
yx
Substitute for “y” into either original equation and solve for “x”…
+
84
15470
154
x
x
yx
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
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.
EXAMPLE : The sum of two numbers is 24. Their difference is 16. Find the two numbers
LINEAR SYSTEMS – Word 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.
EXAMPLE : The sum of two numbers is 24. Their difference is 16. Find the two numbers
24 yx
LINEAR SYSTEMS – Word 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.
EXAMPLE : The sum of two numbers is 24. Their difference is 16. Find the two numbers
16
24
yx
yx
LINEAR SYSTEMS – Word Problems
EXAMPLE : The sum of two numbers is 24. Their difference is 16. Find the two numbers
402
16
24
x
yx
yx
+Addition method…
LINEAR SYSTEMS – Word Problems
EXAMPLE : The sum of two numbers is 24. Their difference is 16. Find the two numbers
20
2
40
2
2
402
16
24
x
x
x
yx
yx
+
Solve for “x”
LINEAR SYSTEMS – Word Problems
EXAMPLE : The sum of two numbers is 24. Their difference is 16. Find the two numbers
20
2
40
2
2
402
16
24
x
x
x
yx
yx
+Substitute for “x” in either equation and solve for “y”…
4
2420
24
y
y
yx
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.
27 yx
LINEAR SYSTEMS – Word Problems
EXAMPLE # 2 : The sum of two numbers is 27. The second number is three more than twice the first number.
32
27
xy
yx
LINEAR SYSTEMS – Word Problems
EXAMPLE # 2 : The sum of two numbers is 27. The second number is three more than twice the first number.
32
27
xy
yx
I will use substitution method…
2732 xx
LINEAR SYSTEMS – Word Problems
EXAMPLE # 2 : The sum of two numbers is 27. The second number is three more than twice the first number.
32
27
xy
yx
Solve for “x”… 2732 xx
24 3
33
2733
2732
x
x
xx
+
8x
LINEAR SYSTEMS – Word Problems
EXAMPLE # 2 : The sum of two numbers is 27. The second number is three more than twice the first number.
32
27
xy
yx
Substitute for “x” and solve for “y’… 2732 xx
24 3
33
2733
2732
x
x
xx
+
8x
19
278
y
y