Solve out the linear equations, Mathematics

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Solve out each of the following equations.

               3( x + 5)= 2 ( -6 - x ) - 2x

Solution

In the given problems we will explained in detail the first problem and the leave most of the explanation out of the following problems.

                               3( x + 5)= 2 ( -6 - x ) - 2x

For this difficulty there are no fractions thus we don't have to worry regarding the first step in the procedure. The next step tells to make simpler both sides.  Thus, we will clear out any parenthesis by multiplying the numbers through & then combine like terms.

                               3(x + 5) = 2 (-6 - x) - 2x

                                  3x+ 15 = -12 - 2x - 2x

                                    3x + 15 = -12 - 4x

The next step is to get all the x's on one side & all the numbers on the other side. Which side the x's go on is up to you and will possibly vary with the problem.  As a rule of thumb usually we will put the variables on the side that will provide a positive coefficient. It is done simply since it is frequently easy to lose track of the minus sign on the coefficient and thus if we ensure it is positive we won't have to worry about it.

Hence, for our case this will mean adding 4x to both of sides and subtracting 15 from both sides.  Notice as well that whereas we will actually put those operations in this time normally we do these operations in our head.

3x+ 15 = -12 - 4 x

3x + 15 -15 + 4x = -12 - 4 x + 4x -15

7 x = -27

 

The next step says to obtain a coefficient of 1 in front of the x.  In this case we can do this by dividing both of sides by a 7.

          7 x/7 = -27/7

            x = - 27 /7

Now, if we've done all of our work accurate x = - 27/ 7 is the solution to the equation.

Then the last & final step is to check the solution.  As pointed out in the procedure outline we have to check the solution in the original equation. It is important, since we may have commit a mistake in the first step and if we did and then checked the answer in the results from that step it might seem to denote that the solution is correct while the reality will be that we don't have the exact answer due to the mistake that we originally made.

The problem of course is that, along with this solution, that verifying might be a little messy.  Let's do it anyway.

 

                24 /7= 24/7               OK

Thus, we did our work properly & the solution to the equation is,

                                x = - 27 /7

Notice that we didn't employ the solution set notation here. For single solutions we will hardly ever do that in this class.  Though, if we had desired to the solution set notation for this problem would be,

                                                             {- 27/7}


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