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Two step equations I


We have already seen how to solve Single Step Equations. These ones seemed almost too simple to use a series of rules or steps to solve.  Time to take it to the next level.

Two Step Equations

A two step equation is one that will require two steps to solve.  They generally have a multiplication/division and a subtraction/addition.  The following are all two step equations.

$2x+5=11$2x+5=11,          $\frac{1}{3}h-3=6$13h3=6,         $19-2j=5$192j=5

Let's look at two methods for solving two step equations.

Method 1 - Backtracking

Two step equations can be set up using a backtracking tool. 

Start by writing the variable (variable) in a box. Then one step at a time mark in the operations that happen in order (according to Order of Operations).  Remember we learnt how to set up equations in backtracking here.  


Solve: $-2x+4=8$2x+4=8

Think: First we need to set up the equation.

  × $-2$2   $+$+$4$4  



Do: Backtrack one step at a time, reversing each operation. 

  × $-2$2   $+$+$4$4  


  × $-2$2   $+$+$4$4  



So $x=-2$x=2.

With all equations we can check our solution. Does $x=-2$x=2 satisfy the equation $-2x+4=8$2x+4=8?


Method 2 - Use formal algebraic techniques

This means using the "do the same to both sides" method, to isolate the $x$x.  That is, get the $x$x on its own.

It is all about reversing the operations. So, we will need to remove the constant (number) term first.  This is done by choosing the reverse operation.


$-2x+4$2x+4 $=$= $8$8 The opposite of addition is subtraction.
$-2x+4-4$2x+44 $=$= $8-4$84 Start by subtracting $4$4 from both sides.
$-2x$2x $=$= $4$4 Simplify both sides of the equation
$\frac{-2x}{-2}$2x2 $=$= $\frac{4}{-2}$42 Then identify the next operation that needs to be reversed. The opposite of multiplying by $-2$2 is dividing by $-2$2.
$x$x $=$= $-2$2 Simplify both sides to find $x$x


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