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$13​h−3=6,         $19-2j=5$19−2j=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 pronumeral (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.  

Example

Solve: $-2x+4=8$−2x+4=8

Think: First we need to set up the equation.

	× $-2$−2		$+$+$4$4
$x$x				$8$8

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

	× $-2$−2		$+$+$4$4
$x$x		$4$4		$8$8

	× $-2$−2		$+$+$4$4
$-2$−2		$4$4		$8$8

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.

$-2x+4=8$−2x+4=8

$-2x+4$−2x+4	$=$=	$8$8	The opposite of addition is subtraction.
$-2x+4-4$−2x+4−4	$=$=	$8-4$8−4	Start by subtracting $4$4 from both sides.
$-2x$−2x	$=$=	$4$4	Simplify both sides of the equation
$\frac{-2x}{-2}$−2x−2​	$=$=	$\frac{4}{-2}$4−2​	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