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

is another way to solve this equation:

      – 2X = a – b

      Let’s divide both sides of the equation by -2

      To make your result more beautiful, you can multiply the numerator and the denominator by – 1.

      The next equation:

      3a – 6X = 6X – 9a

      You can see that on the left side of the equation you have -6X

      and on the right side +6X. It is more convenient for you to have a + in front of X, therefore, you leave +6X on the right side and get rid of -6X on the left side of the equation.

      Add 6X to both sides of the equation:

      3a – 6X +6X = 6X – 9a +6X then 3a = 12X – 9a

      Add 9a to both sides of the equation:

      3a +9a = 12X – 9a +9a then 12a = 12X

      X = a

      Do not read any more until you perform some exercises.

      Practice 3. Solving the equations

      Answers: page 20

      Solutions: page 27

      (Solve for X)

      1. 1 – X = 5 – a

      2. 1 – 2X = X – 4

      3. a – 3X = b – X

      4. 2a – 4X = 2X – 4a

      5. 4b – 2X = 2X – 4b

      6. ab + aX = 2aX + ac

      7. ab + aX = 2aX – ac

      Let us continue and discuss the equation: aX – bX = a – b

      Factor out X which is a common factor for binomial aX – bX,

      then you will get X (a – b) = a – b

      Divide each part of the equation by a – b

      X (a – b) / (a – b) = (a – b) / (a – b)

      X = 1

      Do not read any more until you perform some exercises.

      Practice 4. Solving the equations

      Answers: page 20

      Solutions: page 30

      (Solve for X)

      1. bX – 2b = aX – 2a

      2. b – 2bX = a – 2aX

      3. aX – bX = 1

      4. aX – bX – cX = 2a – 2b – 2c

      5. 3abX – 5a = 3acX +13a

      6. aX – bX = ac – bc

      7. 9a – 4X = 5a – 2X

      8. X – aX = 2 – 2a

      9. aX – bX = b – a

      Let us continue and solve the equation: aX – bX = 2b – 2a

      Factor out X from the left side of the equation.

      X (a – b) = 2b – 2a

      Factor out 2 from the right side of the equation.

      X (a – b) = 2 (b – a) then divide both sides by (a – b)

      Factor out (-1) from the denominator

      Or you can simplify this algebraic expression by factoring out (-1) from the numerator

      Practice 5. Solving the equations

      Answers: page 20

      Solutions: page 32

      Solve for X.

      1. 5aX – 5bX = 10b – 10a

      2. aX – bX – cX = c + b – a

      3. 2X – 3aX = 6a – 4

      4. 3aX – 9bX = 27b – 9a

      5. 4bX – cX = 8c – 32b

      6. abX – acX = ac – ab

      7. X/2 – aX = 1 – 2a

      8. aX/5 +2a = 5a – 4aX

      You can find answers in appendix 1 and solutions in appendix 2.

      2. SYSTEM OF EQUATIONS

      Look at the equation X + Y = 3. X and Y are unknown. You cańt find neither X nor Y from this equation. You need additional information about the “relationship” between them. Such information may be included in an additional equation. For example: X – Y = – 1. Now you have a system of 2 equations:

      1. X + Y = 3

      X – Y = -1

      There are several ways to solve it. The first way: solve for X in any equation, for example, the first one.

      In order to do that, subtract Y from each side of the equation:

      X + Y – Y = 3 – Y; Find X

      X = 3 – Y

      Then put (3 – Y) in place of X in the second equation (X – Y = -1).

      You will obtain: 3 – Y – Y = -1 or 3 – 2Y = -1

      Now solve that equation for Y. Add 2Y to the both sides of the equation.

      3 – 2Y +2Y = -1 +2Y

      3 = -1 +2Y

      Add 1 to the both sides of the equation.

      3 +1 = -1 +1 +2Y

      4 = 2Y