# Solving Equations

## The rules of Algebraic Manipulation

An equation may contain some x-terms and some numbers. We want to manipulate the equation to isolate a single x term on one side and a number on the other side i.e x=some number. The last two sections showed that correct algebraic manipulation maintains balance between the left hand side (LHS) and right hands, and these rules were determined:

You can add a quantity on the LHS so long as you add the same quantity to the RHS

## Subtraction

You can subtract a quantity from the LHS so long as you add the same quantity to the RHS. Note that subtraction is really the addition of a negative number, this is no really seperate rule.

## Multiplication

You can multiply all terms on both sides by a quantity.

## Division

You can divide all terms on both sides by a quantity. Note that since division is really multiplication by a fraction, this is not really a seperate rule.

## Solving Equations using Algebra Solution Interface

The "Solution Interface" will let you perform algebraic operations to solve equations. You are advised to start with 2 steps problems. A solution with the least number of steps can be seem by pressing "Answer". The 3 step problem is slightly harder than the 2 step problem. The 4 step problem contains fractions, and you need to find a strategy to deal with the fractions: try a few times before see the solution after pressing "Answer".

Press for a

exercise

### Exercise: Two Step Problem

Solve for x by using the buttons:

Multiply all terms

Divide all terms

Press to see the answer and working out:

## Solving Equations on Paper

After enough practice on the "Solution Interface" you should be confident. To check your progress solve the equations below and check compare your working out with solution. You may think a 4 step problem is a lot, actually problems in science and engineer can involve several hundred steps. Equations that suggested the existence of anti-matter, the trajectory of rockets to reach outer space, and many other problems, have all been solved on paper. The aim of this website is to give you a deep and visual understanding of maths topics so that you can solve problems without the aid of a calculator or computer.

### Exercise: Two Step Problem on Paper

Solve the following equation:

x input

Numerator

Denominator

Check you answer and working out below:

### Exercise: Three Step Problem on Paper

Solve the following equation:

x input

Numerator

Denominator

Check you answer and working out below:

### Exercise: Four Step Problem on Paper

Solve the following equation:

x input

Numerator

Denominator

Check you answer and working out below:

#### Summary

The approach for solving an equation is to use correct algebraic steps so that one side of the equation contains only a single "x" and the other side contains a number. We have discussed algebra from first principles and focused on the solution of some relatively simple problems. I hope that you have understood the basis of algebra and also the importance of solving equations neatly and systematically on paper; you know have a very good base to understand more advanced algebra and applications of algebra.