solve linear equation w/many operations

Solve:   x - 6 = -12       Solve:   x + 6 = -12       Solve:   x/6 = -12       Solve:   6x = -12 Solve:   - x = -12       Solve:   5x/6 = -12       Solve:   3 + 2x = - 1 Solve:   3 + 2x = 4x - 1      Solve:   3x + 4(x -1) = 2      Solve:   4x + 2 - 3x + 5 - 2(x + 5) = (2x - 5) + (3x + 4) consecutive integer expressions, equations       Solving Equation on Graphing Calculators Intro to Solving Linear Equations                   One Operation Equations   More Than One Operation Equations           Solve Linear Equation w/Many Operations  Solutions to a Linear Equation & Solving Linear Equations Graphically Linear Equation Solver - The Page Does The Work

Intro

      Equations come in many "flavors."   At this point, you should know how to solve the first 8 equations listed below.   This page deals with how to solve equation 9, which is often difficult for the student lacking confidence, and equation 10.

      Use the above links to review the material if you are not able to solve equations 1 through 8.

Solve equations which require undoing 1 operation

1.) x - 6 = -12

2.) x + 6 = -12

3.) x/6 = -12

4.) 6x = -12

5.) - x = -12

6.) 5x/6 = -12

Solve equations which require undoing 2 operations

7.) 3 + 2x = - 1

8.) 3x + 4(x -1) = 2

Solve equations which require undoing many operations, some on each side.

9.) 3 + 2x = 4x - 1

10.) 4x + 2 - 3x + 5 - 2(x + 5) = (2x - 5) + (3x + 4)

      Equation 10 is about as messy as a linear equation gets. Notice that the equivalent equation "-x - 3 = 5x - 1, " in the middle of the problem, is the same in format at equation 9, "3 + 2x = 4x - 1."

      Notice in equation 10, that until you get to the equation "-x - 3 = 5x - 1, " all that's been done is combining like terms.

10.)   Solve: 4x + 2 - 3x + 5 - 2(x + 5) = (2x - 5) + (3x + 4) 4x + 2 - 3x + 5  - 2x - 10 =   2x - 5 +   3x + 4 4x - 3x - 2x + 5 + 2 - 10  =  2x + 3x - 5 + 4         - x             - 3           =       5x       - 1   -x - 3 = 5x - 1 is the line above recopied +x     = +x          -3   = 6x - 1    +1   =    + 1    - 2   = 6x    - 2/6 = 6x/6    - 1/3 = x, the answer

 

      After the equation "-x - 3 = 5x - 1," terms are combined from one side to the other using the following rules.

•       IT DOES NOT MATTER WHICH SIDE YOU CHOOSE. Pick a side on which to collect or put the variables. It is easier to put the variables on the side which already has the most variables.
   
•       DON'T CHANGE THE SIGNS WHEN MOVING TERMS ON THE SAME SIDE, just combine like terms as usual.
   
•       DO CHANGE THE SIGNS WHEN MOVING TERMS FROM ONE SIDE TO THE OTHER because you need to undo the addition or subtraction.

      Here's the solution of the equation "-x - 3 = 5x - 1." The equation has been solved twice showing that it doesn't matter which on which side the variables are collected.

      Put the variables on one side. Put the constants on the other.

10.)   Solve: -x - 3 = 5x - 1 +x     = +x          -3   = 6x - 1    +1   =    + 1    - 2   = 6x    - 2/6 = 6x/6    - 1/3 = x, the answer

10.)   Solve: -x - 3 = 5x - 1 -5x     = -5x         -6x - 3  =     - 1         +3   =    + 3    - 6x = +2    (- 6x)/(-6) = (+2)/(-6)    x = -1/3, the answer

 

 

 

Solve:   3 + 2x = 4x - 1

      Illustrated below is that it doesn't matter on which side the variable is placed.

9.)   Solve: 3 + 2x = 4x - 1      -2x = -2x       3         = 2x - 1  +1   =        + 1     4   = 2x     4/2 = 2x/2     2 = x, the answer

9.)   Solve: 3 + 2x = 4x - 1       -4x = -4x         3 - 2x  =     - 1   -3         =    - 3        - 2x = -4    (- 2x)/(-2) = (-4)/(-2)    x = 2, the answer

 

      Again, the steps in solving an equation like "ax +b = cx + d."

•       Pick a side on which to collect or put the variables.
   
•       DON'T CHANGE THE SIGNS WHEN MOVING TERMS ON THE SAME SIDE, just combine like terms as usual.
   
•       DO CHANGE THE SIGNS WHEN MOVING TERMS FROM ONE SIDE TO THE OTHER because you need to undo the addition or subtraction.

 

 

Problems

Solve & show work.

11.)     3(x+1)+2(x-3)=9

Answer 3(x+1)+2(x-3)=9   3x+3+2x-6=9   5x - 3 = 9      +3   + 3 _________________    5x = 12   5x/5 = 12/5     x = 12/5

 

12.)     4x + 2 = 5x - 3

Answer 4x + 2 = 5x - 3 -4x    -4x ________________    2 = x - 3    +3 =   + 3 ________________ 5 = x

 

13.)     2x + 5 = 7x - 2

Answer 2x + 5 = 7x - 2 -2x     -2x _____________________     5 = 5x - 2     +2 =   + 2 _____________________     +7 = 5x   +7/5 = 5x/5   +7/5 = x

 

14.)     24 - 5x = 4x + 6

Answer 24 - 5x = 4x + 6     +5x  +5x _____________________ 24     = 9x + 6 -6     =   -6 _____________________     +18 = 9x   18/9 = 9x/9       2 = x

 

15.)     4x - 5 = 4x + 1

Answer 4x - 5 = 4x + 1 -4x      -4x _____________________ -5 = 1, always false, so, there is no solution

 

16.)     4(x+1) + 3x + 2 = 7x + 6

Answer 4(x+1) + 3x + 2 = 7x + 6 4x + 4 + 3x + 2 = 7x + 6 7x + 6 = 7x + 6 -7x       -7x ______________ 6 = 6, always true,so, all numbers are solutions

 

 

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