The Easiest Way to Understand Algebra. Algebra equations with answers and solutions - Sergey D. Skudaev

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X + k = 1 + k

      8. X – ab = a – ab

      9. X + c = c – b

      10. X – 2a = a – ab

      11. X + cb = 3cb – c

      12. X – 5 + a = 2a – 5

      13. X +3 – k = 6 – 3k

      14. X – 1 – ab = ab – 1

      15. X – a – b = a – b

      16. X +2a – 3c = 3a – 2c

      You can find the answers in appendix 1. If your answer is wrong try again.

      If you can’t get the right answer read the solution in appendix 2.

      Let’s solve the equation

      4X – 5 = 15

      You can apply the first rule.

      4X – 5 +5 = 15 +5 then 4X = 20.

      How can you find X? You can apply the fourth rule.

      If you divide both sides of an equation by the same numbers, this equation will still be true.

      4X / 4 = 20 / 4, then X = 5.

      To solve equation

      aX – b = c

      Apply the first rule.

      aX – b + b = c + b,

      then aX = c + b

      Now apply the fourth rule.

      If aX = c + b, then aX / a = (c + b) / a

      and X = (c + b) / a

      Do not read any more until you perform some exercises.

      Practice 2. Solving the equations

      Answers: page 19

      Solutions: page 24

      Solve for X:

      1. 2X – 3 = 5

      2. 3X – 5 = 4

      3. 5X +6 = 36

      4. 8X – 5 = 43

      5. 7X – 2 = 19

      6. 4X +8 = 20

      7. 6X – a = 2a

      8. 2X + b = 13b

      9. 7X +3a = a + b

      10. 4X – 2a = 4 +2a

      11. 4X – 3a = a

      12. 3X – 2b = 6 – 14b

      13. 6X – 2a = 24b – 20a

      14. aX – 3a = ab – 2a

      15. 2aX + ab = 2a – ab

      16. 3aX – c = 3ac – 7c

      Answers are in appendix 1.

      Solutions are in appendix 2.

      If you have such an equation to solve:

      X/a – 5 = 6

      Then apply the first rule:

      X/a – 5 +5 = 6 +5

      X/a = 6 +5

      X/a = 11

      Then apply the third rule.

      X/a * a = 11 *a

      X = 11a

      Let’s solve the equation:

      2X – 4b = 2bc

      Apply the first rule:

      2X – 4b +4b = 2bc +4b,

      then 2X = 2bc +4b

      Apply the third rule:

      2X/ 2 = (2bc +4b) / 2

      You should know how to divide a binomial by a monomial.

      If you have forgotten it, you could find the rule by yourself. Can you write

      (2bc+4b) /2 = 2bc/2 +4b/2? Yes, you can.

      Let us check. Suppose, c = 2 and b = 3.

      To divide a binomial by 2, try to divide each monomial by 2

      2*3*2/2 +4*3/2 = 12

      Now try to solve the binomial first and then divide by 2

      (2*3*2 +4*3) /2 then 24/2 = 12

      We got the same answer. It means that

      (a+ b) /2 = a/2 + b/2.

      We discovered a rule: To divide a binomial by a number, divide each monomial

      inside the binomial by that number. Come back to your equation.

      2X = 2bc +4b. Then

      2X / 2 = 2bc / 2 +4b / 2

      Then X = bc +2b

      You can factor out b and get

      X = b (c +2)

      Whenever you don’t know the rule, you can put any numbers in place of the letters and check equality. Discover rules by yourself.

      Let us solve a more complicated equation:

      Multiply both sides of the equation by 5X.

      5X – 5 = 50X

      Use the 2nd rule, subtract 5X from both sides:

      5X – 5 – 5X = 50X – 5X

      – 5 = 45X

      or 45X = – 5

      Divide both sides by 45

      45X/45 = -5/45

      X = – 1/9

      The next equation:

      Find the common denominator:

      Since + aX – aX = 0, our equation becomes simple:

      Use the 3rd rule: multiply both sides of the equation by (a + b)

      Then b – bX = c (a + b)

      Apply the 2nd rule, subtract b from both sides of the equation:

      b – bX – b =c (a + b) – b

      Then -bX =c (a + b) -b Divide both sides by b:

      To make this algebraic expression more beautiful, multiply the numerator and denominator by (-1).

      You can do that because (-1) / (-1) = 1. If you multiply any number by 1 the number will not be changed.

      Then

      The next equation:

      – 2X = a – b

      It

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