Answer:
x = -3
Step-by-step explanation:
You are asked to solve the 3-step linear equation -6x -4 = -10x -16.
<h3>Solution</h3>
<u>Step 1</u> is collect the variable terms on one side of the equal sign. You may find it convenient to add the opposite of the variable term that has the lowest coefficient.
-6x -4 +10x = -10x -16 +10x . . . . . . add 10x to both sides
4x -4 = -16 . . . . . . . . . . . . . . . simplify
<u>Step 2</u> is to isolate the variable term by adding the opposite of any constant term on the same side of the equal sign.
4x -4 +4 = -16 +4 . . . . . . . . add 4 to both sides
4x = -12 . . . . . . . . . . . . simplify
<u>Step 3</u> is to isolate the variable by dividing by its coefficient.
(4x)/4 = (-12)/4 . . . . . . . divide both sides by 4
x = -3 . . . . . . . . . . . . simplify
The value of x is -3.
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<em>Additional comments</em>
<u>Avoiding errors</u>
By choosing to add 10x rather than 6x in step 1, we ensure the coefficient of x is positive. Having a positive coefficient for the variable tends to reduce errors.
If we added 6x instead, the equation going into step 3 would be 12 = -4x, so we would have x = 12/-4. Dividing by negative numbers sometimes causes confusion, so we try to avoid it.
We notice after step 1 that all of the numbers in the equation are divisible by 4, the coefficient of x. We could divide by 4 at that point to get ...
x -1 = -4
Then adding 1 gives the solution:
x = -3
In short, the steps described above offer guidance to suggest what you can do if you don't see a simpler way.
<u>Alternate solution steps</u>
Another way any linear equation can be solved is to ...
- subtract one side to give an equation in "general form":
something = 0. - divide by the coefficient of x
- add the opposite of the remaining constant
Again, choosing to subtract the side of the equation with the lowest x-coefficient can be helpful in the end.
-6x -4 -(-10x -16) = 0 . . . . . . subtract the right side from both sides
4x +12 = 0 . . . . . simplify
x +3 = 0 . . . . . . . divide by the x-coefficient
x = -3 . . . . . . . . . add the opposite of the constant
<u>Rule of equality</u>
It is extremely important to understand that <em>whatever you do to one side of the equation must also be done to the other side</em>. When we say "add -3" we always mean "add -3 to both sides of the equation." (This is the foundation of Algebra, and is what allows you to solve equations of all kinds.)
