Two Step Equations I

People running backwards
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In this lesson we look at solving two step equations using back tracking flowcharts.

A typical two step equation is 2N + 5 = 11

We can use “Trial and Error”, (also called “Guess Check and Improve”), to work out the number value of “N” which gives us an answer of 11.

Trial and Error

Trial and Error, or guessing answers, is fine for simple equations.

But when we have more difficult equations like 7k – 110 = 2, it could take many guesses until we get the answer.

Consequently, we generally use a different method for solving two step equations which is called “Flowcharting” or “Back Tracking”.

Here are the main steps involved with Back Tracking using Flowcharts:

B Track steps
The first step involves working out the order of the operations that are applied to our letter variable.

The “Order of Operations” is called “PEMDAS” in the USA, and “BODMAS” in Australia. Other countries have very similar sets of rules.

BODMAS

In 2N + 5 = 11 there is adding of 5, and also multiplying by 2.

The Order of Operations tells us that multiplying by 2 happens before the adding of 5.

Example A

To make a Forward Flowchart to represent “N”, and the multiplying by 2 followed by the adding of 5, we need to draw three rectangles.

In between the rectangles we place two arrows, because this equation has two operations (x2 and +5) performed in it.

The resulting Forward Flowchart looks like this:

Flowchart A

Here is a partially completed Flowchart for an equation that has multiplying and subtraction in it.

See if you can fill in the missing items.

Flowchart Q1

(Answer: The Flowchart requires x3 then -2 and a 3k in the middle rectangle box).

Here is another example of a two step equation, but this one has dividing and then adding.

See if you can fill in the missing items.

Flowchart Q2

(Answer: The Flowchart requires on the arrows divided by 5 or /5, followed
by + 2. x/5 needs to go in the middle box).

Here is a third example Flowchart for you to complete.

Flowchart Q3

This equation has “Brackets” (or “Parenthesis”) in it, and whatever operation is inside these must be done first.

The required operations are – 3 then x 2, and not the usual x 2 then -3.

This happens because the brackets force us to do the – 3 first, whereas without brackets we would do – 3 last.

(Answer: The Flowchart requires on the arrows -3 then x 2, and the final box needs to contain the left hand side of the original equation which was 2(a-3).

In every Two Step Equation Forward Flowchart:

The letter variable always goes on its own in the first box

The whole left hand side of the equation always goes in the third and final box.

The Operations that were done on the variable letter go onto the arrows, in BODMAS or PEMDAS order.

Back Tracking Flowcharts

Let’s now look at how we can add more boxes and arrows onto our Forward Flowchart, to make a full “Back Tracking Flowchart”.  This flowchart will solve the equation for us.

We add extra boxes under each of our Forward boxes, and also add back tracking arrows onto our diagram.

Whenever we make a Back Tracking Flowchart, we always create the same overall structure.

Backtrack A

This may look very complicated at first, but things will become much clearer when we do some number examples.

The following “Opposite” operations are always used for solving equations.

Backtrack Opposites

Let’s return to our first example equation: 2N + 5 = 11, and see how its solution of N=3 can be worked out using Back Tracking.

Backtrack A

Checking Solutions Using Substitution

We can always check the solution to any equation by substituting the number answer we obtained back into the original equation.

Here is how we can check the N=3 solution for the equation 2N + 5 = 11.

Sub Check A

Back Tracking Examples

Here are some examples for you to try.

Simply complete the missing items in the Back Tracking Flowchart.

You can check your work against the answers that are given in the section which follows these three examples.

Backtrack Q1

This next question has divided by in it. Note that we write divided by 5 as /5 when applying division to letter variables.

Backtrack Q2

This final example has Brackets in it, which must be done first, regardless of what operation is inside them.

Baqcktrack Q3

Answers for Back Tracking Examples

k = (10 + 2) / 3 = 12 / 3 = 4 Middle Box has 12 in it, and left side box has 4 in it. Final Answer is k=4.

x = (6-2) x 5 = 4×5 = 20 Middle box has 4 in it and left side box has 20 in it. Final Answer is x=20.

a = 8 / 2 + 3 = 4 + 3 = 7 Middle box has (a-3) in it. Top arrow has x 2.

Bottom arrows have divided by 2 and + 3 going backwards right to left.

Bottom middle box has 4 in it. Final Answer in left bottom box is 7, so a=7.

Note that it is okay to replace the / symbol with the “divided by” symbol in your answers for any of these equations, except for letter variables which must be written as x/5, y/3, 2m/5, etc.

Video About Back Tracking

Here is a video about using Back Tracking to solve for unknown values. It flows along quite quickly, so you may need to pause it every now and then to review what is happening.

Back Tracking Two Step Equations Summary

We need to set up three double box rectangles, with arrows in between them.

The letter variable always goes on its own in the top left hand box

The whole left hand “Algebra” side of the equation always goes in the third and final top box.

The Right hand side number for the equation always goes in the very right hand bottom box.

The Operations that were done on the variable letter go onto the arrows, in BODMAS or PEMDAS order on the top, and in Opposite order on the bottom.

The number answer always ends up in the bottom left hand side box.

The flowcharts for two step equations are always set up using the structure shown in the following diagram.

Backtracking Summary

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