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Chapter 11
Solving Linear Equations. Integer
Problems
c H. Feiner 2011
11.1
Basic Terminology
Consecutive Integers: 7, 8, 9 are consecutive integers. x, x + 1, x + 2 are consecutive integers.
Consecutive Even Integers: 8, 10, 12 are consecutive even integers. x, x + 2, x + 4 are consecutive even
integers.
Consecutive Odd Integers: 7, 9, 11 are consecutive odd integers. x, x + 2, x + 4 are consecutive even integers.
11.2
Examples
Some of the problems here are simple. The solution can be worked out fast by quick reasoning. The benefit
of these problems is not to find the solution by reasoning, but by learning algebraic steps applicable to more
challenging situations.
The general method is not to read a problem first and understand it. Many of my colleagues will disagree
with me. I propose creating a preamble in which you write a symbol or set of symbols for each element in
the problem.
Look at the end of the problem which asks the question. Let x be the number we are looking for. Write this
as the first step in the preamble. Start reading the problem and develop the preamble step by step. Write
the symbol(s) for each step on a new line. Finished converting all the steps in the problem into symbols.
Now look at your preamble and understand it. It will be easier to obtain an equation using the symbols from
the preamble. Then solve the equation by the method introduced this far for linear equations. (A linear
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CHAPTER 11. SOLVING LINEAR EQUATIONS. INTEGER PROBLEMS
equation has a variable to the first degree (exponent).)
Example 1:
The sum of three consecutive positive integers is 705. Find the integers.
Solution:
(Method 1)
Preamble:
Let x be the smallest of the three consecutive integers (think of some number, like x = 20).
Then x + 1 is the middle number (like 20 + 1 = 21).
And x + 2 is the largest number (like 20 + 2 = 22).
The sum of three integers is x + (x + 1) + (x + 2) = 3x + 3.
Equation:
Sum of integers
3x + 3
3x + 3 − 3
3x
=
=
=
=
sum of integers
705
705 − 3
702
3x
3
=
702
3
x =
234
Smallest integer: x = 234
Middle integer: x + 1 = 234 + 1 = 235
largest integer: x + 2 = 234 + 2 = 236
(Method 2)
Preamble:
Let x be the middle of the three consecutive integers (think of some number, like x = 20).
Then x − 1 is the smallest number (like 20 − 1 = 19).
And x + 1 is the largest number (like 20 + 1 = 21).
The sum of three integers is (x − 1) + x + (x + 1) = 3x.
Equation:
Sum of integers =
3x =
3x
=
3
x =
sum of integers
705
705
3
235
Smallest integer: x − 1 = 235 − 1 = 234
Middle integer: x = 235
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11.2. EXAMPLES
largest integer: x + 1 = 235 + 1 = 236
(Method 3)
Preamble:
Let x be the largest of the three consecutive integers (think of some number, like x = 20).
Then x − 1 is the middle number (like 20 − 1 = 19).
And x − 2 is the largest number (like 20 − 2 = 18).
The sum of three integers is (x − 2) + (x − 1) + x = 3x − 3.
Equation:
Sum of integers
3x − 3
3x − 3 + 3
3x
3x
3
=
=
=
=
=
x =
sum of integers
705
705 + 3
708
708
3
236
Smallest integer: x − 2 = 236 − 2 = 234
Middle integer: x − 1 = 236 − 1 = 235
largest integer: x = 236
Example 2:
Find two integers whose sum is 82.
11 more than three times the smaller number is the same as 18 less than twice the larger number.
Find the numbers.
Solution:
(Method 1)
Preamble:
Let x be the smaller number (like x = 20).
Then the larger number is 82 − x (like 82-20=62).
11 more than three times the smaller number: 3x + 11 (like 3(20) + 11).
18 less than twice the larger number: 2(82 − x) − 18 = 164 − 2x − 18 = 146 − 2x (like 2(82 − 20) − 18).
Equation:
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CHAPTER 11. SOLVING LINEAR EQUATIONS. INTEGER PROBLEMS
11 more than three times the smaller number
3x + 11
3x + 2x + 11
5x + 11
5x + 11 − 11
5x
5
x
5
x
=
=
=
=
=
=
18 less than twice the larger number
146 − 2x
146 − 2x + 2x
146
146 − 11
135
135
=
5
= 27
The smaller number is x = 27.
The larger number is 82 − 27 = 55.
(Method 2)
Preamble:
Let x be the larger number (like x = 50).
Then the smaller number is 82 − x (like 82-50=62).
11 more than three times the smaller number: 3(82 − x) + 11 = 3(82) − 3x + 11 = 246 + 11 − 3x = 257 − 3x
(like 3(82 − 50) + 11).
18 less than twice the larger number: 2x − 18 (like 2(20) − 18).
Equation:
11 more than three times the smaller number
257 − 3x
257 − 3x + 3x
257
257 + 18
275
5x
5
x
5
x
=
=
=
=
=
=
=
18 less than twice the larger number
2x − 18
2x + 3x − 18
5x − 18
5x − 18 + 18
5x
275
275
=
5
= 55
The larger number is 55. The smaller number is 82 − 55 = 27.
Example 3:
Find two consecutive odd integers such that 60 less than three times the larger number equals 73 more than
twice the smaller number.
Solution:
(Method 1)
Preamble:
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11.3. EXERCISES 11
Let x be the smaller odd number (like x = 21).
Then the larger number is x + 2 (like 21 + 2 = 23).
60 less than three times the larger number: 3(x + 2) − 60 = 3x + 6 − 60 = 3x − 54 (like 3(21 + 2) − 60).
73 more than twice the smaller number: 2x + 73 (like 2(21) + 73).
Equation:
60 less than three times the larger number
3x − 54
3x − 54 + 54
3x
3x − 2x
x
=
=
=
=
=
=
73 more than twice the smaller number
2x + 73
2x + 73 + 54
2x + 127
2x − 2x + 127
127
The smaller number is x = 127.
The larger number is 127 + 2 = 129.
(Method 2)
Preamble:
Let x be the larger odd number (like x = 21).
Then the smaller number is x − 2 (like 21 − 2 = 19).
60 less than three times the larger number: 3x − 60 (like 3(21) − 60).
73 more than twice the smaller number: 2(x − 2) + 73 = 2x − 4 + 73 = 2x + 69 (like 2(21 − 2) + 73).
Equation:
60 less than three times the larger number
3x − 60
3x − 60 + 60
3x
3x − 2x
x
=
=
=
=
=
=
73 more than twice the smaller number
2x + 69
2x + 69 + 60
2x + 129
2x − 2x + 129
129
The larger number is 129. The smaller number is 129 − 2.
11.3
Exercises 11
1. The sum of three consecutive positive integers is 1, 128. Find the integers.
2. Find two consecutive even integers such that 150 less than three times the smaller number equals 148
more than twice the larger number.
3. Find two integers whose sum is 425.
10 more than six times the smaller number is the same twice the larger number.
Find the numbers.
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CHAPTER 11. SOLVING LINEAR EQUATIONS. INTEGER PROBLEMS
STOP!
1. The sum of three consecutive positive integers is 1, 128. Find the integers.
Solution:
Preamble:
Let x be the smallest of the three consecutive integers (think of some number, like x = 20).
Then x + 1 is the middle number (like 20 + 1 = 21).
And x + 2 is the largest number (like 20 + 2 = 22).
The sum of three integers is x + (x + 1) + (x + 2) = 3x + 3.
Equation:
Sum of integers
3x + 3
3x + 3 − 3
3x
3x
3
x
=
=
=
=
=
=
sum of integers
1, 128
1, 128 − 3
1, 125
1, 125
3
375
Smallest integer: x = 375
Middle integer: x + 1 = 375 + 1 = 376
largest integer: x + 2 = 375 + 2 = 377
2. Find two consecutive even integers such that 150 less than three times the smaller number equals 148
more than twice the larger number.
Solution:
Preamble:
Let x be the smaller odd number (like x = 21).
Then the larger number is x + 2 (like 21 + 2 = 23).
150 less than three times the smaller number: 3x − 150 (like 3(21) − 150).
148 more than twice the larger number 2(x + 2) + 148 = 2x + 4 + 148 = 2x + 152 (like 2(21 + 2) + 148).
Equation:
150 less than three times the smaller number
3x − 150
3x − 2x − 150
x − 150
x − 150 + 150
x
=
=
=
=
=
=
148 more than twice the larger number
2x + 152
2x − 2x + 152
152
152 + 150
302
The smaller number is x = 302.
The larger number is 302 + 2 = 304.
3. Find two integers whose sum is 425.
10 more than six times the smaller number is the same twice the larger number.
Find the numbers.
Solution:
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11.3. EXERCISES 11
Preamble:
Let x be the smaller number (like x = 20).
Then the larger number is 425 − x (like 425-20=405).
10 more than six times the smaller number: 6x + 10 (like 6(20) + 10).
twice the larger number: 2(425 − x) = 850 − 2x (like 2(425 − 20)).
Equation:
10 more than six times the smaller number
6x + 10
6x + 2x + 10
8x + 10
8x + 10 − 10
8x
The smaller number is x = 105.
The larger number is 425 − 105 = 320.
=
=
=
=
=
=
twice the larger number
850 − 2x
850 − 2x + 2x
850
850 − 10
840
8
x =
8
840
8
x =
105