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2.2: Adding Integers - Mathematics


Like our work with the whole numbers, addition of integers is best explained through the use of number line diagrams. However, before we start, let’s take a moment to discuss the concept of a vector.

Vectors

A vector is a mathematical object that possesses two import qualities: (1) magnitude or length, and (2) directon.

Vectors are a fundamental problem solving tool in mathematics, science, and engineering. In physics, vectors are used to represent forces, position, velocity, and acceleration, while engineers use vectors to represent both internal and external forces on structures, such as bridges and buildings. In this course, and in this particular section, we will concentrate on the use of vectors to help explain addition of integers.

Vectors on the Number Line

Consider the number line in Figure (PageIndex{1}).

Above the line we’ve drawn a vector with tail starting at the integer 0 and arrowhead finishing at the integer 4. There are two important things to note about this vector:

  1. The magnitude (length) of the vector in Figure (PageIndex{1}) is four.
  2. The vector in Figure (PageIndex{1}) points to the right.

We will agree that the vector in Figure (PageIndex{1}) represents positive four. It is not important that the vector start at the origin. Consider, for example, the vector pictured in Figure (PageIndex{2}).

Again, there are two important observations to be made:

  1. The magnitude (length) of the vector in Figure (PageIndex{2}) is four.
  2. The vector in Figure (PageIndex{2}) points to the right.

Hopefully, you have the idea. Any vector that has length 4 and points to the right will represent positive four, regardless of its starting or finishing point. Conversely, consider the vector in Figure (PageIndex{3}), which starts at the integer 4 and finishes at the integer −3.

Two observations:

  1. The magnitude (length) of the vector in Figure (PageIndex{3}) is seven.
  2. The vector in Figure (PageIndex{3}) points to the left.

We will agree that the vector in Figure (PageIndex{3}) represents negative seven. We could select different starting and finishing points for our vector, but as long as the vector has length seven and points to the left, it represents the integer −7.

Important Observation

A vector pointing to the right represents a positive number. A vector pointing to the left represents a negative number.

Magnitude and Absolute Value

In Figure (PageIndex{1}) and Figure (PageIndex{2}), the vectors pictured represent the integer positive four. Note that the absolute value of four is four; that is, |4| = 4. Note also that this absolute value is the magnitude or length of the vectors representing the integer positive four in Figure (PageIndex{1}) and Figure (PageIndex{2}).

In Figure (PageIndex{3}), the vector pictured represents the integer −7. Note that | − 7| = 7. This shows that the absolute value represents the magnitude or length of the vector representing −7.

Magnitude and Absolute Value

If a is an integer, then |a| gives the magnitude or length of the vector that represents the integer a.

Adding Integers with Like Signs

Because the positive integers are also whole numbers, we’ve already seen how to add to them in Section 1.2.

Example 1

Find the sum 3 + 4.

Solution

To add the positive integers 3 and 4, proceed as follows.

  1. Start at the integer 0, then draw a vector 3 units in length pointing to the right, as shown in Figure (PageIndex{4}). This arrow has magnitude (length) three and represents the positive integer 3.
  2. Draw a second vector of length four that points to the right, starting at the end of the first vector representing the positive integer 3. This arrow has magnitude (length) four and represents the positive integer 4.
  3. The sum of the positive integers 3 and 4 could be represented by a vector that starts at the integer 0 and ends at the positive integer 7. However, we prefer to mark this sum on the number line as a solid dot at the positive integer 7. This integer represents the sum of the positive integers 3 and 4.

Thus, 3 + 4 = 7

Exercise

Use a number line diagram to show the sum 5 + 7.

Answer

12

Negative integers are added in a similar fashion.

Example 2

Find the sum −3+(−4).

Solution

To add the negative integers −3 and −4, proceed as follows.

  1. Start at the integer 0, then draw a vector 3 units in length pointing to the left, as shown in Figure (PageIndex{5}). This arrow has magnitude (length) three and represents the negative integer −3.
  2. Draw a second vector of length four that points to the left, starting at the end of the first vector representing the negative integer −3. This arrow has magnitude (length) four and represents the negative integer −4.
  3. The sum of the negative integers −3 and −4 could be represented by a vector that starts at the integer 0 and ends at the negative integer −7. However, we prefer to mark this sum on the number line as a solid dot at the negative integer −7. This integer represents the sum of the negative integers −3 and −4.

Thus, −3+(−4) = −7.

Drawing on Physical Intuition. Imagine that you are “walking the number line” in Figure (PageIndex{5}). You start at the origin (zero) and take 3 paces to the left. Next, you walk an additional four paces to the left, landing at the number −7

Exercise

Use a number line diagram to show the sum −7+(−3).

Answer

−10

It should come as no surprise that the procedure used to add two negative integers comprises two steps.

Adding Two Negative Integers

To add two negative integers, proceed as follows:

  1. Add the magnitudes of the integers.
  2. Prefix the common negative sign

Example 3

Find the sums: (a) −4+(−5), (b) −12 + (−9), and (c) −2+(−16).

Solution

We’ll examine three separate but equivalent approaches, as discussed in the narrative above.

a) The number line schematic

shows that (−4) + (−5) = −9.

b) Drawing on physical intuition, start at zero, walk 12 units to the left, then an additional 9 units to the left. You should find yourself 21 units to the left of the origin (zero). Hence, −12 + (−9) = −21.

c) Following the algorithm above in “Adding Two Negative Integers,” first add the magnitudes of −2 and −16; that is, 2+16 = 18. Now prefix the common sign. Hence, −2+(−16) = −18.

Exercise

Find the sum: −5+(−9).

Answer

−14

Adding Integers with Unlike Signs

Adding integers with unlike signs is no harder than adding integers with like signs.

Example 4

Find the sum −8 + 4.

Solution

To find the sum −8 + 4, proceed as follows:

  1. Start at the integer 0, then draw a vector eight units in length pointing to the left, as shown in Figure (PageIndex{6}). This arrow has magnitude (length) eight and represents the negative integer −8.
  2. Draw a second vector of length four that points to the right, starting at the end of the first vector representing the negative integer −8. This arrow (also shown in Figure (PageIndex{6})) has magnitude (length) four and represents the positive integer 4.
  3. The sum of the negative integers −8 and 4 could be represented by a vector that starts at the integer 0 and ends at the negative integer −4. However, we prefer to mark this sum on the number line as a solid dot at the negative integer −4. This integer represents the sum of the integers −8 and 4.

Thus, −8+4= −4.

Drawing on Physical Intuition. Imagine you are “walking the number line in Figure (PageIndex{6}). You start at the origin (zero) and walk eight paces to the left. Next, turn around and walk four paces to the right, landing on the number −4.

Exercise

Use a number line diagram to show the sum −9 + 2.

Answer

−7

Note that adding integers with unlike signs is a subtractive process. This is due to the reversal of direction experienced in drawing Figure (PageIndex{6}) in Example 4.

Adding Two Integers with Unlike Signs

To add two integers with unlike signs, proceed as follows:

  1. Subtract the smaller magnitude from the larger magnitude.
  2. Prefix the sign of the number with the larger magnitude.

For example, to find the sum −8 + 4 of Example 4, we would note that the integers −8 and 4 have magnitudes 8 and 4, respectively. We would then apply the process outlined in “Adding Two Integers with Unlike Signs.”

  1. Subtract the smaller magnitude from the larger magnitude; that is, 8 − 4 = 4.
  2. Prefix the sign of the number with the larger magnitude. Because −8 has the larger magnitude and its sign is negative, we prefix a negative sign to the difference of the magnitudes. Thus, −8+4= −4.

Example 5

Find the sums: (a) 5+ (−8), (b) −12+ 16, and (c) −117+115.

Solution

We’ll examine three separate but equivalent approaches, as discussed in the narrative above.

a) The number line schematic

shows that 5 + (−8) = −3.

b) Drawing on physical intuition, start at zero, walk 12 units to the left, then turn around and walk 16 units to the right. You should find yourself 4 units to the right of the origin (zero). Hence, −12 + 16 = 4.

c) Following the algorithm in “Adding Two Integers with Unlike Signs,” subtract the smaller magnitude from the larger magnitude, thus 117−115 = 2. Because −117 has the larger magnitude and its sign is negative, we prefix a negative sign to the difference of the magnitudes. Thus, −117 + 115 = −2.

Exercise

Use a number line diagram to show the sum 5 + (−11).

Answer

−6

Properties of Addition of Integers

You will be pleased to learn that the properties of addition for whole numbers also apply to addition of integers.

The Commutative Property of Addition

Let a and b represent two integers. Then,

[a + b = b + a. onumber ]

Example 6

Show that 5 + (−7) = −7 + 5.

Solution

The number line schematic

shows that 5 + (−7) = −2. On the other hand, the number line schematic

shows that −7+5= −2. Therefore, 5 + (−7) = −7 + 5.

Exercise

Use a number line diagram to show that −8 + 6 is the same as 6 + (−8).

Addition of integers is also associative.

The Associative Property of Addition

Let a, b, and c represent integers. Then,

[(a + b) + c = a + (b + c) onumber ].

Example 7

Show that (−9 + 6) + 2 = −9 + (6 + 2).

Solution

On the left, the grouping symbols demand that we add −9 and 6 first. Thus,

[ egin{aligned} (−9 + 6) + 2 & = −3+2 ~ & = −1. end{aligned} onumber ]

On the right, the grouping symbols demand that we add 6 and 2 first. Thus,

[ egin{aligned} −9 + (6 + 2) & = −9+8 ~ & = −1. end{aligned} onumber ]

Both sides simplify to −1. Therefore, (−9 + 6) + 2 = −9 + (6 + 2).

Exercise

Show that the expression (−8 + 5) + 3 is the same as −8 + (5 + 3) by simplifying each of the two expressions independently.

The Additive Identity Property

The integer zero is called the additive identity. If a is any integer, then

a +0= a and 0 + a = a.

Thus, for example, −8+0= −8 and 0 + (−113) = −113.

Finally, every integer has a unique opposite, called its additive inverse.

The Additive Inverse Property

Let a represent any integer. Then there is a unique integer −a, called the opposite or additive inverse of a, such that

a + (−a) = 0 and −a + a = 0.

Example 8

Show that 5 + (−5) = 0.

Solution

The number line schematic

clearly shows that 5 + (−5) = 0.

Exercise

Use a number line diagram to show that 9 + (−9) = 0.

Important Observation

We have used several equivalent phrases to pronounce the integer −a. We’ve used “the opposite of a,” ”negative a,” and “the additive inverse of a.” All are equivalent pronunciations.

Grouping for Efficiency

Order of operations require that we perform all additions as they occur, working from left to right.

Example 9

Simplify −7+8+(−9) + 12.

Solution

We perform the additions as they occur, working left to right.

[ egin{aligned} =7 + 8 + (-9) +12 = 1 + (-9) +12 ~ & extcolor{red}{ ext{ Working left to right, } -7 + 8 = 1.} = -8 + 12 ~ & extcolor{red}{ ext{ Working left to right, } 1 + (-9_ = -8.} =4 ~ & extcolor{red}{ ~ -8 + 12 = 4} end{aligned} onumber ]

Thus, −7+8+(−9) + 12 = 4.

Exercise

Simplify: −8+9+(−4) + 2.

Answer

−1

The commutative property of addition tells us that changing the order of addition does not change the answer. The associative property of addition tells us that a sum is not affected by regrouping. Let’s work Example 9 again, first grouping positive and negative numbers together.

Example 10

Simplify −7+8+(−9) + 12.

Solution

The commutative and associative properties allows us to change the order of addition and regroup.

[ egin{aligned} -7 + 8 + (-9) + 12 = -7 + (-9) + 8 + 12 ~ & extcolor{red}{ ext{ Use the commutative property to change the order.}} = [-7+(-9)] + [8+12] ~ & extcolor{red}{ ext{ Use the associative property to regroup.}} =-16 + 20 ~ & extcolor{red}{ ext{ Add the negatives. Add the positives.}} =4 ~ & extcolor{red}{ ext{ One final addition.}} end{aligned} onumber ]

Thus, −7+8+(−9) + 12 = 4.

Exercise

Simplify: −11 + 7 + (−12) + 3.

Answer

−13

At first glance, there seems to be no advantage in using the technique in Example 10 over the technique used in Example 9. However, the technique in Example 10 is much quicker in practice, particularly if you eliminate some of the explanatory steps.

Efficient Grouping

When asked to find the sum of a number of integers, it is most efficient to first add all the positive integers, then add the negatives, then add the results.

Example 11

Simplify −7+8+(−9) + 12.

Solution

Add the positive integers first, then the negatives, then add the results.

[ egin{aligned} -7 + 8 + (-9) +12 = 20 + (-16) ~ & extcolor{red}{ ext{ Add the positives: } 8 + 12 = 20.} ~ & extcolor{red}{ ext{ Add the negatives: } -7 + (-9) = -16.} =4 ~ & extcolor{red}{ ext{ Add the results: } 20 + (-16) = 4.} end{aligned} onumber ]

Thus, −7+8+(−9) + 12 = 4.

Exercise

Simplify: −11 + 3 + (−2) + 7.

Answer

−7

Using Correct Notation

Never write +−! That is, the notation

9 +−4 and − 8 +−6

should not be used. Instead, use grouping symbols as follows:

9+(−4) and −8+(−6)

Exercises

In Exercises 1-12, what integer is represented by the given vector?

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.


In Exercises 13-36, find the sum of the given integers.

13. −15 + 1

14. −1 + 18

15. 18 + (−10)

16. 2+(−19)

17. −10 + (−12)

18. −1+(−7)

19. 5 + 10

20. 1 + 12

21. 2+5

22. 14 + 1

23. 19 + (−15)

24. 20 + (−17)

25. −2+(−7)

26. −14 + (−6)

27. −6 + 16

28. −2 + 14

29. −11 + (−6)

30. −7+(−8)

31. 14 + (−9)

32. 5+(−15)

33. 10 + 11

34. 14 + 11

35. −13 + 1

36. −8+2


In Exercises 37-52, state the property of addition depicted by the given identity.

37. −1 + (3 + (−8)) = (−1 + 3) + (−8)

38. −4 + (6 + (−5)) = (−4 + 6) + (−5)

39. 7+(−7) = 0

40. 14 + (−14) = 0

41. 15 + (−18) = −18 + 15

42. 14 + (−8) = −8 + 14

43. −15 + 0 = −15

44. −11 + 0 = −11

45. −7 + (1 + (−6)) = (−7 + 1) + (−6)

46. −4 + (8 + (−1)) = (−4 + 8) + (−1)

47. 17 + (−2) = −2 + 17

48. 5+(−13) = −13 + 5

49. −4+0= −4

50. −7+0= −7

51. 19 + (−19) = 0

52. 5+(−5) = 0


In Exercises 53-64, state the additive inverse of the given integer.

53. 18

54. 10

55. 12

56. 15

57. −16

58. −4

59. 11

60. 13

61. −15

62. −19

63. −18

64. −9


In Exercises 65-80, find the sum of the given integers.

65. 6+(−1) + 3 + (−4)

66. 6+(−3) + 2 + (−7)

67. 15 + (−1) + 2

68. 11 + (−16) + 16

69. −17 + 12 + 3

70. −5+(−3) + 2

71. 7 + 20 + 19

72. 14 + (−14) + (−20)

73. 4+(−8) + 2 + (−5)

74. 6+(−3) + 7 + (−2)

75. 7+(−8) + 2 + (−1)

76. 8+(−9) + 5 + (−3)

77. 9+(−3) + 4 + (−1)

78. 1+(−9) + 7 + (−6)

79. 9 + 10 + 2

80. −6 + 15 + (−18)


81. Bank Account. Gerry opened a new bank account, depositing a check for $215. He then made several withdraws of $40, $75, and $20 before depositing another check for $185. How much is in Gerry’s account now?

82. Dead Sea Sinking. Due to tectonic plate movement, the Dead Sea is sinking about 1 meter each year. If it’s currently −418 meters now, what will Dead Sea elevation be in 5 years? Write an expression that models this situation and compute the result.

83. Profit and Loss. Profits and losses for the first six months of the fiscal year for a small business are shown in the following bar chart. Sum the profits and losses from each month. Was there a net profit or loss over the six-month period? How much?

84. Was there a net profit or loss over the six-month period? How much?


Answers

1. 4

3. 6

5. −5

7. −6

9. 10

11. −7

13. −14

15. 8

17. −22

19. 15

21. 7

23. 4

25. −9

27. 10

29. −17

31. 5

33. 21

35. −12

37. Associative property of addition

39. Additive inverse property

41. Commutative property of addition

43. Additive identity property

45. Associative property of addition

47. Commutative property of addition

49. Additive identity property

51. Additive inverse property

53. −18

55. −12

57. 16

59. −11

61. 15

63. 18

65. 4

67. 16

69. −2

71. 46

73. −7

75. 0

77. 9

79. 21

81. $265

83. Net Profit: $24,000


Related Resources

The various resources listed below are aligned to the same standard, (7NS01) taken from the CCSM (Common Core Standards For Mathematics) as the Integers Worksheet shown above.

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers represent addition and subtraction on a horizontal or vertical number line diagram.

  • Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
  • Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
  • Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
  • Apply properties of operations as strategies to add and subtract rational numbers.

Example/Guidance

Worksheet

Similar to the above listing, the resources below are aligned to related standards in the Common Core For Mathematics that together support the following learning outcome:

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers


Adding and subtracting integers

You already know how to add 3 + 4 and so on. But there are many ways to add integers. One way to add integers is by using a number line.

We always start at zero. Our first number is negative four (-4) so we move 4 units to the left. We then have plus negative three (-3) which is the same as subtracting 3 so we move 3 more units to the left. This gives us the value of negative seven, (-7).

We do the same thing if we have a positive integer, but instead we move to the right.

You can also add integers and variables.

When subtracting something from something we wish to find out the difference between the two numbers. When you subtract a negative number from any number the difference is even bigger. The distance from the seabed at a depth of 150ft and an airplane flying at 3000ft altitude at sea level is

$3000-(-150) = 3000 + 150 = 3150ft.$

Thus when we subtract negative numbers, we get:

Subtracting −3 is the same as adding 3.

If we have a plus sign before the parentheses then we will not change the signs within the parentheses

If we have a minus sign before the parentheses then we the signs within the parentheses will change.

Two negatives make one positive!


Floating-Point Numbers or Floats in Python

Floats get assigned the same way as int values in Python. Here is how that looks in code:

If you run this code in your integrated development environment, you will get the following output:

Again, because we assigned a float value to our variable, it becomes a float literal. The same rule applies to large floating-decimal point values as integers in that you have to use underscores to delineate them versus commas. Another thing to know is that you can also assign values to a float by using exponential notation – or E notation for short. Here is an example of how that looks:

In this example, we created a variable named lotto and assigned it the value of a one with six 0’s following it, or, 1000000.0. When we create variables with E notation, those values are always floats.


Adding and subtracting with integers

Adding can make less and subtraction can make more

  1. Calculate each of the following:
    1. (10 + 4 + (-4))
    2. (10 + (-4) + 4)
    3. (3 + 8 + (-8))
    4. (3 + (-8) + 8)

    The numbers 1 2 3 4 etc. that we use to count, are called natural numbers.

    Natural numbers can be arranged in any order to add and subtract them. This is also the case for integers.

    1. Calculate each of the following:
      1. (18 + 12)
      2. (12 + 18)
      3. ( 2 + 4 + 6)
      4. (6 + 4 + 2)
      5. (2 + 6 + 4)
      6. (4 + 2 + 6)
      7. ( 4 + 6 + 2)
      8. ( 6 + 2 + 4)
      9. ( 6 + (-2) + 4)
      10. ( 4 + 6 + (-2))
      11. (4 + (-2) + 6)
      12. ( (-2) + 4 + 6)
      13. ( 6 + 4 + (-2))
      14. ( (-2) + 6 + 4)
      15. ( (-6) + 4 + 2)
      1. ( (-5) + 10)
      2. ( 10 + (-5))
      3. ( (-8) + 20)
      4. ( 20 - 8)
      5. ( 30 + (-10))
      6. (30 + (-20))
      7. (30 + (-30))
      8. ( 10 + (-5) + (-3))
      9. ((-5) + 7 + (-3) + 5)
      10. ( (-5) + 2 + (-7) + 4)
      1. 20 + (an unknown number) = 50
      2. 50 + (an unknown number) = 20
      3. 20 + (an unknown number) = 10
      4. (an unknown number) + (-25) = 50
      5. (an unknown number) + (-25) = -50

      Statements like these are also called number sentences.

      An incomplete number sentence, where some numbers are not known at first, is sometimes called an open number sentence:

      A closed number sentence is where all the numbers are known:

      1. Use the idea of additive inverses to explain why each of the following statements is true:
        1. ( 43 + (-50) = -7)
        2. ( 60+ (-85) = -25 )
        1. (80 + (-60))
        2. (500 + (-200) + (-200))
        1. Is (100 + (-20) + (-20) = 60), or does it equal something else?
        2. What do you think ((-20) + (-20)) is equal to?
        1. (20 - 20)
        2. (50 - 20)
        3. ( (-20) - (-20))
        4. ((-50) - (-20))
        1. ( 20 - (-10))
        2. ( 100 - (-100))
        3. ( 20 + (-10))
        4. (100 + (-100))
        5. ((-20) - (-10))
        6. ( (-100) - (-100))
        7. ( (-20) + (-10))
        8. ( (-100) + (-100))
        1. Subtracting a positive number from a negative number has the same effect as dding the additive inverse of the positive number.
        2. Adding a negative number to a positive number has the same effect as adding the additive inverse of the negative number.
        3. Subtracting a negative number from a positive number has the same effect as subtracting the additive inverse of the negative number.
        4. Adding a negative number to a positive number has the same effect as subtracting the additive inverse of the negative number.
        5. Adding a positive number to a negative number has the same effect as adding the additive inverse of the positive number.
        6. Adding a positive number to a negative number has the same effect as subtracting the additive inverse of the positive number.
        7. Subtracting a positive number from a negative number has the same effect as subtracting the additive inverse of the positive number.
        8. Subtracting a negative number from a positive number has the same effect as adding the additive inverse of the negative number.

        Comparing integers and solving problems

        1. Fill <, > or = into the block to make the relationship between the numbers true:
          1. -103 ☐ -99
          2. -699 ☐ -701
          3. 30 ☐ -30
          4. 10-7 ☐ -(10-7)
          5. -121 ☐ -200
          6. -12 - 5 ☐ -(12 + 5)
          7. -199 ☐ -110

          Subtracting Integers

          Why is (-4) – 4 not 0, instead -8? How are the integers added despite the subtraction sign in between? Hammer home such interesting facts and solve subtraction problems with integers in the range -20 to 20.

          Levitate your integer subtraction skills by performing the appropriate operation on integers between -100 and 100, and obtaining the difference.


          If a number has a fractional part, you can make the number change to be the closest, next integer value. This is called rounding. Rounding the number 6.78 will make it be 7 and rounding 9.3 will give you 9 . If a number has a fractional part greater than or equal to 0.5 , the number will round up to the next whole integer value with the higher value. Otherwise, it will round down to the next lowest integer value.

          For negative numbers, they round toward the absolute value (the absolute value of -8 is 8 ) of the number. So, -5.23 rounds to -5 and -2.68 rounds to -3 .


          How to Find the Greatest Common Divisor of Two Integers

          wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, 40 people, some anonymous, worked to edit and improve it over time.

          This article has been viewed 569,023 times.

          The Greatest Common Divisor (GCD) of two whole numbers, also called the Greatest Common Factor (GCF) and the Highest Common Factor (HCF), is the largest whole number that's a divisor (factor) of both of them. For instance, the largest number that divides into both 20 and 16 is 4. (Both 16 and 20 have larger factors, but no larger common factors -- for instance, 8 is a factor of 16, but it's not a factor of 20.) In grade school, most people are taught a "guess-and-check" method of finding the GCD. Instead, there is a simple and systematic way of doing this that always leads to the correct answer. The method is called "Euclid's algorithm." If you want to know how to truly find the Greatest Common Divisor of two integers, see Step 1 to get started. [1] X Research source


          Two-step equations

          The main difference between one-step equationsand two-step equations is that one more step you need to do in order to solve a two-step equation. That additional step may be something like multiplying the variable by a certain number to get rid of a fraction in front of it. Otherwise, the rules are the same as before and these equations are just as easy to learn and solve as are the one-step ones

          An example of two-step equations

          The easiest way to learn math is through the use of examples and some practice. So, we are going to explain the process of solving these equations on this example:

          The first thing you need to do here is get all the variables on one side (in this case – the left side) and all the numbers on the other. Like this:

          After that, you have to add those two numbers. You should get this as a result:

          This concluded the first step. What is left now is to get rid of the number next to the variable. You can do that by dividing the whole equation with the number 4. Like this:

          That was the second step. You found the value of the variable and the equation is now solved. I am sure you will all agree this was not hard to do. And although these equations can be a bit more complicated, the principle is always the same. First you need to simplify the equation by moving the numbers around (and do not forget to change signs when you move numbers from side to side)(buy cialis online) and then get rid of the number in front of the variable to find its value (by multiplying or dividing the number with itself). Of course, you can perform these steps in reverse (getting rid of the number in front of the variable first, then the “moving around” bit), but this way will save you some time and effort in most cases.
          If you want to practice solving two-step equations, you can download the worksheets below this article for free.


          Watch the video: Adding Integers (October 2021).