# How to solve this linear equation? which has an x on each side

$5x + 8 = 10x + \dfrac{3}{6}$

And I have achieved this result:

$x = 9$

Is my result correct?

I have already posted two other questions related to this topic, I'm a programmer and am learning Math out of my interest, this is not homework.

My steps taken:

10x - 5x = 48 - 3

5x = 45

5x/5 = 45/5

x = 9

This is my steps which led to a wrong answer.

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mathematicsi.com/rearranging-equations may be a useful reference about equation manipulation. –  JB King Jul 1 at 9:30
Perhaps if you posted the steps you took to get this wrong answer we can see where you are going wrong. Remember with an equation whatever you do to one side of the equation you must do to the other. Get all the $x$ terms on one side of the equation and everything else on the other then simplify. –  Warren Hill Jul 1 at 9:53
In the steps zou took, you forgot to multiply the left side with $6$. You should get $10x-5x = 8-3/6$ and then multiply that by $6$ to get $30x = 45$, not $5x=45$. Also, it is better if you replace $3/6$ with $1/2$ and just multiply by $2$. –  5xum Jul 1 at 11:01
At a higher level, the mistake was attempting to combine two steps, grouping the x-terms on one side and multiplying by 6 to get rid of the fraction. It those steps had been done separately they would probably have been done correctly. –  Patricia Shanahan Jul 1 at 13:55

$5x + 8 = 10x + 3/6$

Let's gather all the x's on one side and the other stuff on the other side as well as reduce the fraction to lowest terms first:

$10x-5x = 8-\frac{1}{2}$

$5x = 8-.5$

$5x = 7.5$

$x = 1.5$ or $x=\frac{3}{2}$ or $x=1+\frac{1}{2}$

Now, to compute each side as a verfication:

$LHS= 5x+8 = 5*\frac{3}{2}+8 = 7.5+8 = 15.5$

$RHS = 10x+\frac{3}{6} = 10*\frac{3}{2}+\frac{1}{2} = 15+0.5 = 15.5$

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It is very easy to see if your result is NOT correct. If $x=9$, then the left side of your equation is

$$5 \cdot 9 + 8 = 45+8=53$$

and the right is $$10\cdot 9 + \frac36 = 90 + \frac12$$ which is not the same.

Hint: When solving the equation, first move all the values containing $x$ on one side, so you get

$$5x-10x = -8+\frac36$$ and solve from here on.

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$$5x+8 = 10x + \frac{3}{6} \overset{(1)}{\iff} 8-\frac{3}{6}=10x-5x \overset{(2)}{\iff} \frac{45}{6} = 5x \overset{(3)}\iff x=\frac{9}{6}\overset{(4)}{=} \frac{3}{2}$$

Steps:

(1) subtract $5x+\frac{3}{6}$ on both sides of the equation

(2) Simplify, note that $8-\frac{3}{6} = \frac{48}{6}-\frac{3}{6} = \frac{45}{6}$

(3) Divide by $5$ (or, equivalently, multiply by $\frac{1}{5}$), note that $\frac{45}{6}\cdot\frac{1}{5} = \frac{9\cdot 5}{6}\cdot\frac{1}{5} = \frac{9}{6}$

(4) Simplify, note that $\frac{9}{6}=\frac{3\cdot 3}{2\cdot 3} = \frac{3}{2}$.

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It's very simple to see whether your solution is correct or not. You can simply take your $x$ and put back into the original equation. Remember, the equal sign ($=$) means that both sides have to be the same and therefore, if both sides are not equivalent then your $x$ value is wrong and you have gone wrong somewhere. $$5 * (9) + 8 \not= 10 * (9) + \frac{3}{6}$$

If you to solve the equation then follow these steps:

1. Simplify both sides as much as possible. For example, get rid of all the fractions
2. Bring all your $x$'s to one side
3. Isolate $x$ by following order of operations BEDMAS/PEDMAS
4. Plug $x$ back into the original equation and see if you are correct

Good luck!

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A good tip would be to put all of the variables on the left hand side of the equation and all of the numbers on the right side equation.

$5x + 8 = 10x + \dfrac{3}{6}$

First we would reduce the fraction

$5x + 8 = 10x + \dfrac{1}{2}$

and then we would subtract $10x$ from both sides, so we will have

$5x-10x + 8 = \dfrac{1}{2}$

Then we would subtract $8$ from both sides and we will have

$5x-10x = \dfrac{1}{2} - 8$

We know that $5x - 10x = -5x$ but what about the right hand side of the equation? We need to multiply $8$ by $\frac{2}{2}$

$5x-10x = \dfrac{1}{2} - 8 \times \dfrac{2}{2}$

Now, we must simplify the terms.

$-5x = \dfrac{1-16}{2}$

$-5x = \dfrac{-15}{2}$

Then, we divide $-5$ from both sides and get

$x = \frac{\dfrac{-15}{2}}{-5}$

Flipping the second fraction, we have

$x = \dfrac{-15}{2}\frac{-1}{5}$

So the final answer is $x = \frac{-15}{10}$ which reduces to $x = \frac{3}{2}$

Now we shall substitute $x = \frac{3}{2}$ back into the original equation

$5 \times \dfrac{3}{2} + 8 = 10 \times \dfrac{3}{2} + \dfrac{1}{2}$

$\dfrac{15}{2}+ 8 = \dfrac{30}{2} + \dfrac{1}{2}$

$\dfrac{15}{2}+ 8 \times \dfrac{2}{2} = \dfrac{30}{2} + \dfrac{1}{2}$

$\dfrac{15+16}{2} = \dfrac{30+1}{2}$

$\dfrac{31}{2}=\dfrac{31}{2}$

Since the left hand side equals the right hand side, we have found our x.

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It looks like your first attempt to solve this went wrong at the first step. By the usual method you would get $10x-5x$ on one side of the equation and $8-\frac{3}{6}$ on the other. You got the $10x-5x$ on one side all right, but for some reason you decided to multiply the other side by a factor of $6$.

A step that multiplies things by $6$ is a reasonable step for this problem, but you have to do it on both sides of the equation at the same time, not just on one side.

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