# Linear Equations And Problem Solving

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

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It is easy to make a mistake here, so make sure that you distribute the number in front of the parentheses to all the terms inside.

Solve: $$3(x 2)-1=x-3(x 1)$$ First, distribute the 3 and –3, and collect like terms.

Let’s look at one more two-step example before we jump up in difficulty again.

Make sure that you understand each step shown and work through the problem as well.

Solve: $$3x=12$$ Since $$x$$ is being multiplied by 3, the plan is to divide by 3 on both sides: $$\begin3x &=12\\ \dfrac &=\dfrac\\ x&= \boxed\end$$ To check our answer, we will let $$x = 4$$ and substitute it back into the equation: $$\begin3x &= 12\\3(4) &= 12 \\ 12 &= 12\end$$ Just as before, since this is a true statement, we know our answer is correct.

In the next example, instead of the variable being multiplied by a value, a value is being subtracted from the variable.

$$\begin 3(x 2)-1 &=x-3(x 1)\ 3x 6-1&=x-3x-3 \ 3x 5&=-2x-3\end$$ Now we can add 2x to both sides.

(Remember you will get the same answer if you instead subtracted 3x from both sides) $$\begin 3x 5\color &=-2x-3\color\ 5x 5& =-3\end$$ From here, we can solve as we did with other two-step equations.

Solve: $$2x-7=13$$ Notice the two operations happening to $$x$$: it is being multiplied by 2 and then having 7 subtracted. But, only the $$x$$ is being multiplied by 2, so the first step will be to add 7 to both sides. Adding 7 to both sides: $$\begin 2x-7 &= 13\ 2x-7 \color & =13 \color\ 2x&=20\end$$ Now divide both sides by 2: $$\begin 2x &=20 \ \dfrac&=\dfrac\ x&= \boxed\end$$ Just like with simpler problems, you can check your answer by substituting your value of $$x$$ back into the original equation.

$$\begin2x-7&=13\ 2(10) – 7 &= 13\ 13 &= 13\end$$ This is true, so we have the correct answer.

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