---
title: "Equations: cheat sheet"
subtitle: "Algebra Foundations"
---

# Equations

> The words and methods from this unit, with an example of each.

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**expression**: A calculation with a letter in it. It has a value once you know the letter, but it does not say that anything is equal.

$$2x + 3$$

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**equation**: Two expressions with = between them: a statement that they are equal, like a balanced scale.

$$2x + 3 = 11$$

_Balanced when x = 4_

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**solution**: The value of the letter that makes the equation true.

$$x = 4$$

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**inverse operations**: Each step has an opposite that undoes it: adding and subtracting, multiplying and dividing.

$$+ \leftrightarrow -\qquad \times \leftrightarrow \div$$

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**check by substitution**: Put the value in place of the letter and work out both sides. If they match, it is a solution. Check every answer in the first equation.

$$2(4) + 3 = 11 \checkmark$$

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**backtracking**: Undo the steps done to $x$ with their opposites, last step first.

$$\begin{aligned} 3x + 7 &= 19 \\ 19 - 7 &= 12 \\ 12 \div 3 &= 4 \end{aligned}$$

_Multiply by 3, then add 7. Undo + 7 first._

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**balance method**: Do the same to both sides (take away, or share into equal groups) until $x$ is on its own.

$$\begin{aligned} 3x + 5 &= 20 \\ 3x &= 15 \\ x &= 5 \end{aligned}$$

_Take 5 from both sides, then share both into 3 groups._

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**x on both sides**: Take the same number of $x$ from both sides first, so the bags are on one side only.

$$\begin{aligned} 5x &= 2x + 12 \\ 3x &= 12 \\ x &= 4 \end{aligned}$$

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