# Finding an equivalent equation by adding x and then [subtracting] 1/2

#### Illvoices

##### Junior Member
So I need to know what this means in linear equations.
1. They asked me to first find the equivalent equation by adding x and then [subtracting] 1/2
1/2+4x=-x+2
1/2+4x+x=-x+2+x
5x+1/2=2
5x+1/2-1/2=2-1/2
5x=3/2

So is this equation equivalent of not and what does that mean.

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#### ksdhart2

##### Full Member
The general rule is that if you perform an operation to one side of an equation you must then also perform the same operation to the other side. When I was in grade school we were taught this with the silly mnemonic device "Whatever you do to grandma, you have to do to Grandpa too" :roll: In this instance, you've performed two operations - "Adding x and then [subtracting] 1/2." Both times you performed an identical operation to both sides of the equation, so you're fine as far as that goes.

But if, for whatever reason, you still find yourself doubting, another approach is to note that two equations are considered equivalent if and only if they have the same solution set. So ask yourself, how many solutions are there to $$\displaystyle 5x = \dfrac{3}{2}$$? And what are they? If you plug the solution(s) into the other equation $$\displaystyle \dfrac{1}{2} + 4x = -x + 2$$, what do you find? Do the solution(s) to $$\displaystyle 5x = \dfrac{3}{2}$$ also satisfy $$\displaystyle \dfrac{1}{2} + 4x = -x + 2$$? Lastly, how many solutions are there to $$\displaystyle \dfrac{1}{2} + 4x = -x + 2$$? Are there any other solutions that haven't already been accounted for? Based on all of this, what can you conclude about whether the equations are equivalent?

#### Jomo

##### Elite Member
So I need to know what this means in linear equations.
1. They asked me to first find the equivalent equation by adding x and then 1/2
1/2+4x=-x+2
1/2+4x+x=-x+2+x
5x+1/2=2
5x+1/2-1/2=2-1/2
5x=3/2

So is this equation equivalent of not and what does that mean.
If you have an equation and then preform the same operations on both sides then the resulting equation and the original equation are said to be equivalent. However, you can not multiply or divide both sides by 0 nor (I guess) can you add 0 to both sides.

So accordingly, you have 5 equivalent equations, namely 1/2+4x=-x+2, 1/2+4x+x=-x+2+x, 5x+1/2=2, 5x+1/2-1/2=2-1/2, 5x=3/2.

Note that you did not have to move closer to getting to solve for x. Another words, you could have added 7 to both sides getting 7 1/2 +4x = -x + 9 and this is also equivalent to the other 3 equations.

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#### mmm4444bot

##### Super Moderator
Staff member
… They asked me to first find the equivalent equation by adding x and then [subtracting] 1/2 …

1/2 + 4x=-x + 2

1/2 + 4x + x = -x + 2 + x

5x + 1/2 = 2

5x + 1/2 - 1/2 = 2 - 1/2

5x = 3/2

So is this equation equivalent [or] not and what does that mean.
All five equations are equivalent. It means that they each have the same solution as the others.

When they said that, they were emphasizing that adding and subtracting those terms won't change the solution even though it creates a new equation; the new equation has the same solution as the original, so it's equivalent.

#### JeffM

##### Elite Member
If you have an equation and then preform the same operations on both sides then the resulting equation and the original equation are said to be equivalent. However, you can not multiply or divide both sides by 0 nor (I guess) can you add 0 to both sides.

So accordingly, you have 5 equivalent equations, namely 1/2+4x=-x+2, 1/2+4x+x=-x+2+x, 5x+1/2=2, 5x+1/2-1/2=2-1/2, 5x=3/2.

Note that you did not have to move closer to getting to solve for x. Another words, you could have added 7 to both sides getting 7 1/2 +4x = -x + 9 and this is also equivalent to the other 3 equations.
You certainly can multiply both sides of an equation by 0: the result, 0 = 0, is true. It just is not useful. It may be useful to add or subtract zero to both sides.