# Equations » Reversed arrow chain method

## Note:

When you have an equation with a variable on both sides of the =-sign, you cannot use the reversed arrow chain method.

When you have something like 3`x` + 5 = 7`x` – 2, check balance method.

When you have something like 5`x`^{2} + 3`x` = 2`x` + 4, check quadratic equations.

## Contents

1. Regular arrow chain2. Reversed arrow chain with linear equations

3. Reversed arrow chain with quadratic equations

## 1. Regular arrow chain

If you want to work with a reversed arrow chain, you have to know how to make a normal arrow chain for a formula. You start with looking at the variable. What are the operations that have to be done with the variable. These operations you put on arrows. Of course in the order that they have to be done according to the formula. What has to be done first, according to the calculation rules will be on the first arrow.

*Examples*

Formula | Arrow chain | ||||||

y = 7x + 3 |
× 7 | + 3 | |||||

x |
… | y |
|||||

b = 3(a + 2) + 4 |
+ 2 | × 3 | + 4 | ||||

a |
… | … | b |

## 2. Reversed arrow chain with linear equation

Linear equations do not have any powers (like squares) in the equation.

### Rules

When you reverse an arrow chain, the following happens to the operators:

× becomes :

+ becomes –

: becomes ×

– becomes +

When you have an equation, you first make the normal arrow chain and then the reversed arrow chain. Calculate the solution with the reversed arrow chain.

*Examples*

Solve the equation: 4x + 12 = 104
Calculation: (104 – 12) : 4 = 23. Answer: x = 23 |
Solve the equations: 4(x + 12) = 104
Calculation: 104 : 4 – 12 = 14. Answer: x = 14 |

### Check

If you want you can check your solutions by filling in your answers in the formula.

Example 1: 4 × 23 + 12 = 92 + 12 = 104 and that is correct!

Example 2: 4 × (14 + 12) = 4 × 26 = 104 and that is correct!

## 3. Reversed arrow chain with quadratic equations

Works the same as with the linear equations except for the following:The arrow for squared | reversed is a double arrow: | |

#^{2} |
||

– |

Because, for example, –5 × –5 is also 25 you have two possible solutions for quadratic equations. When you have to find the square root of a negative number the equation has no solutions.

*Example 1*

Solve the equation: `x`^{2} + 5 = 41

Arrow chain will be: |
#^{2} |
+ 5 | ||||

x |
… | 41 | ||||

Reversed arrow chain will be: |
x = x = – |
… | – 5 |
41 |

Calculation: 41 – 5 = 36 en = 6

Answer: `x` = 6 of `x` = –6

Check: 6^{2} + 5 = 36 + 5 = 41

Check: (–6)^{2} + 5 = 36 + 5 = 41

*Example 2*

Solve the equation: `a`^{2} – 5 = –10

Arrow chain will be: |
#^{2} |
– 5 | ||||

a |
… | –10 | ||||

Reversed arrow chain will be: |
a = a = – |
… | + 5 |
–10 |

Calculation: –10 + 5 = –5 and = not possible.

Answer: no solutions