When we have equations that start off with variable letters or brackets on both sides, we need to do some extra steps as part of solving them.
The reason we might want to solve equations like this, is to find the common “y coordinate” of the intersection point of two lines.
It can also be used to find when two items starting at different points will meet up with each other.
Letters both sides methods can also be used to compare the Economy (Cost versus Distance) of cars running on different types of fuels, or with different engine sizes.
In a Business, the method is used to find the position where enough sales dollars are generated, to cover the costs of running the Business.
This sales and cost balancing process is called establishing the “Break Even Point”, and is used a lot in Accounting.
In this lesson we look at how to solve equations which have variable letters and/or Brackets on both sides of the Equation.
Prior to doing this lesson, you need to know how to solve basic equations, as well as the “Distributive Rule” for expanding brackets.
If you need to review these concepts, then check out our previous lessons at the following links:
Variable Letter On Both Sides Equation
Here is what a basic letters both sides situation looks like.
Before we can solve equations like this, we have to do some extra steps to remove the letter term from one of the sides.
This involves applying either SUBTRACTION OR ADDITION of a variable letter term value on both sides to the equation.
The steps we need to do are as follows.
Note that by identifying the smaller letter item and using that, it greatly helps reducing the prescence of negative signed items to deal with.
We can in fact reverse out and eliminate either letter item, but our work is easier if we eliminate the smaller item.
Solving Letter On Both Sides Equations
Here is how we apply the standard steps to solve a letters both sides equation.
Once we have completed getting the letter variable on one side, we can now use either “Onion Skin”, “Flowcharts”, or “Algebra Reversing”, to finish solving the equation.
Here is how to finish the question using the “Onion Skins” method:
If you are not familiar with Solving Equations using the “Onion Skins”, then you can review our lesson on this at the following link:
Letters Both Sides Videos
This video from Mahalo Maths explains one example equation fully.
Here is another Video Example which has a fraction answer.
The following video has some fractions to deal with on both sides.
We deal with them by multiplying by the lowest common multiple.
This is a reasonably involved example, which takes a bit of working out:
Variables Both Sides Example
Variables on both sides do not always involve subtraction as the first step.
Note that -5h is smaller than -3h, because -5 is smaller than -3, and so we ADD to both sides to eliminate the -5h.
We can now finish off solving our equation using Onion Skins
Brackets Both Sides Equations
Sometimes questions have the letter term on both sides, and also have some brackets which we need to expand.
If you are not familiar with Expanding Brackets using the “Distributive Rule”, then you can review our lesson on this at the following link:
The following example shows how we FIRST EXPAND THE BRACKETS, and then after this perform the usual letter on both sides step.
Once we have completed expanding all brackets and getting the letter variable on one side, we can now use either “Onion Skin”, “Flowcharts”, or “Algebra Reversing”, to finish solving the equation.
Here is how to finish the question using the “Onion Skins” method:
Videos for Brackets on Both Sides
In this video, there are some brackets to expand using the distributive rule, and then after this we do letters both sides working out.
Here is another video about solving equations with brackets on both sides.
Here are the steps we always need to follow when solving equations with brackets and/or letter variables on both sides.
It is possible to occasionally get some strange answers for our letter variables on both sides questions, where either every number is an answer, or there is no number answer.
The following web page has some good examples of these abnormal types:
Worksheets on Equations
The following worksheet has a good mixture of questions and includes brackets questions:
Here is another worksheet with similar types of questions to Worksheet 1
This third worksheet lets you click to get the answers, and also allows you to generate a number of different questions to work on.
Online Equations Quiz
The following Online Quiz also has worked solutions.
Note you need to page back using your web browswer back button after viewing a solution, to get back to the quiz.
Click the link below to do the Quiz
Solving Equations Calculator
The following web page has an awesome calculator where you can type in your brackets and/or variable on both sides equation, and it will then show you a fully worked algebra solution that gets the answer:
The current Stats at the bottom of the Solution, indicate that this calculator has solved over 39 million equations since it first went onto the web in 2004!
Click the link below to use this Calculator / Solver
Here at Passy’s world we love the above Equations Solver App, because it shows the Algebra the computer performed, as well as giving the final answer.
People who are able to take mathematical items like this, and turn them into Apps are really in demand at the moment, and hopefully in the future students will do more Applied Mathematics using Apps that are freely available.
They will do this instead of learning so much Algebra pen and paper manual working out for equations.
Solving Equations Millionaire Game
This free Online Game is a Millionaire type game, where the later questions have variables on both sides.
Play it at the following link:
Expanding Brackets Using Distributive Rule
How to Translate Word Problems into Equations
Solving One Step Addition Equations
Solving One Step Subtraction Equations
Solving One Step Equations – All Types
Two Step Equations Using Flowcharts and Back Tracking
Two Step Equations Using Reverse Operations
Three Step Equations Using Flowcharts and Back Tracking
Balance Beam Equations
Solving Equations Using Onion Skin Methods
Solving Equations Word Problems
Transposing and Rearranging Formula Equations
eBay Problem Solved Using Algebra Equations
Microsoft Mathematics Equations Solver
Real World Mathematics Formulas and Equations
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