Having trouble with a linear equation.

The equation I have been given is:

$5n+7 = 31-3n$

I keep coming up with $n = 12$ which makes sense because I was told in the following example to subtract the like terms.


Which is of course $x = -4$.

So I am being told that $n = 3$ which I can understand why it would when you ADD the like terms in the first equation but I was told to subtract. Am I wrong? or is this answer wrong - and if I am wrong what rule am I missing that would have me add like terms instead?

  • 2
    $\begingroup$ Adding $3n$ to both sides is the same as subtracting $-3n$ from both sides. But, just like you, I prefer to add. $\endgroup$ – André Nicolas Mar 31 '15 at 5:23

Bunch up like terms on either side of the equation, so you want the terms containing $n$ to be all on one side, so you get

\begin{align} 5n+7&=31-3n\\ \Longrightarrow 5n+(3n)&=31-7\\ \Longrightarrow 8n&=24. \end{align} You can easily solve it from there.

Note. If they said subtract, then they may mean subtract $-3n$ from both sides, which of course means to add $3n$ since $-(-3n)=3n$.

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    $\begingroup$ Ah, entirely makes sense. I overlooked that it was -3n which to cancel out needs +3n. Because if I took 3n from -3n I'd be left with -6n. Thank you. $\endgroup$ – Jesse Mar 31 '15 at 5:26

$$\begin{array} {cc} 5n+7 = 31-3n & \text{Add } 3n \text{ to both sides} \\ 5n + 7 + 3n = 31 - 3n + 3n & \\ 8n + 7 = 31 & \text{Subtract } 7 \text{ from both sides} \\ 8n + 7 - 7 = 31 - 7& \\ 8n = 24 & \text{Divide both sides by } 8 \\ 8n \div 8 = 24 \div 8 & \\ n = 3 & \\ \end{array}$$


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