Rearrangement

[1123581321]

hi,

Was wondering how we can be sure when rearranging an equation/formula, that the worked out formula gives us the correct answer, since the formula is the basis for our interpretation...

[ewmon]

1. Manipulate an equation by treating both sides of it identically. Thus, add 2 to both sides, or divide both sides by v², or take the square root of both sides, etc.

 

2. When you manipulated it as desired, for example, d = ... , plug values into the right side of the equation to determine the value of the left side.

 

3. Then substitute the value of the left side for where the left side appears in the original equation along with the other values being used. Verify that the original equation holds true with these values.

 

For example, begin with a² + 2x = n and solve for x. Subtract a² from both sides, giving 2x = n – a². Divide both sides by 2, giving x = (n – a²)/2. Then plug in values into the right side of the equation. So, for n = 19 and a = 3, then x = (19 – 3²)/2 = (19 – 9)/2 = 10/2 = 5. Finally, go back to the original equation, a² + 2x = n, and plug in all the values. 3² +2×5 = 19, giving 9 + 10 = 19 which is 19 = 19.

 

If all the values are not known, these steps may produce something looking like 19p + w³ = 19p + w³, which still proves that both sides of the equation were correctly manipulated (that is, manipulated identically) in Step 1.

 

If plugging values back into the original equation produces two sides of the equation that are not identical, then the an error occurred somewhere in Steps 1, 2 and/or 3.

[Mr Skeptic]

You make sure that if two things were equal before your change, they remain equal afterward. One way is to do the exact same thing to both sides, whether it be multiply, add, take logarithm, etc. Another is to replace one thing by something equal, for example if x = 3y + 5, you can replace an x in one side by the other value that is equal to it. Another thing you can do is multiply just one side by 1, or add 0 to just one side. This last bit might seem silly but remember that there are lots of things equal to 0 or 1.

[khaled]

it's true that if x = 3y + 5 then,

 

1. change sides

x = 3y + 5

y = x/3 - 5/3

 

2. validation

x = 3(x/3 - 5/3) + 5 = 3x/3 - 3*5/3 + 5 = x -5+5

y = (3y + 5)/3 - 5/3 = 3y/3 + 5/3 - 5/3 = y +5-5 = y