Problem Solving Involving Linear Equations

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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.

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.

\(\begin5x 5\color &=-3\color\\ 5x &=-8\\ \dfrac&=\dfrac\\ x &= \dfrac \\ &=\boxed\end\) This was a tough one, so remember to check your answer and make sure no mistake was made.

To do that, you will be making sure that the following is a true statement: \(3\left(-\dfrac 2\right)-1=\left(-\dfrac\right)-3\left(-\dfrac 1\right)\) (Note: it does work – but you have to be really careful about parentheses!

To check this, verify the following is true: \(\begin4x &= 8\ 4(2) &= 8 \ 8 &= 8\end\) This is a true statement, so our answer is correct.

Let’s try a couple more examples before moving on to more complex equations.

Solve: \(3x 2=4x-1\) Since both sides are simplified (there are no parentheses we need to figure out and no like terms to combine), the next step is to get all of the x’s on one side of the equation and all the numbers on the other side.

The same rule applies – whatever you do to one side of the equation, you must do to the other side as well!

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