# Linear Equation Problem Solving

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Since you are undoing the operations to $$x$$, you will work from the “outside in”.

This is easier to understand when you see it in an example.

It is possible to either move the $$3x$$ or the $$4x$$. Since it is positive, you would do this by subtracting it from both sides: $$\begin3x 2 &=4x-1\\ 3x 2\color &=4x-1\color\\ -x 2 & =-1\end$$ Now the equation looks like those that were worked before.

The next step is to subtract 2 from both sides: $$\begin-x 2\color &= -1\color\\-x=-3\end$$ Finally, since $$-x= -1x$$ (this is always true), divide both sides by $$-1$$: $$\begin\dfrac &=\dfrac\\ x&=3\end$$ You should take a moment and verify that the following is a true statement: $$3(3) 2 = 4(3) – 1$$ In the next example, we will need to use the distributive property before solving.

In each case, the steps will be to first simplify both sides, then use what we have been doing to isolate the variable.

We will first take an in depth look at an example to see how this all works.

$$\begin5w 2 &= 9\ 5w 2 \color &= 9 \color\ 5w &= 7\ \dfrac &=\dfrac\w=\boxed\end$$ The fraction on the right can’t be simplified, so that is our final answer. Then: $$\begin5w 2 &= 9\ 5\left(\dfrac\right) 2 &= 9\ 7 2 &= 9\ 9 &= 9 \end$$ So, we have the correct answer once again!

In the following examples, there are more variable terms and possibly some simplification that needs to take place.

(That’s why equations like these are often called “one-step” equations) Anytime you are solving linear equations, you can always check your answer by substituting it back into the equation.

If you get a true statement, then the answer is correct.

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