Linear Equation Problem Solving

Linear Equation Problem Solving-44
Solving linear equations in algebra is done with multiplication, division, or reciprocals.

Solving linear equations in algebra is done with multiplication, division, or reciprocals.

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In these equations, we will need to undo two operations in order to isolate the variable.

In each of the examples above, there was a single step to perform before we had our answer.

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.

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!

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.

This isn’t 100% necessary for every problem, but it is a good habit so we will do it for our equations.

In this example, our original equation was \(4x = 8\).

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