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Lesson 1: Algebraic equations basics

# Intro to equations

Learn what an equation is and what it means to find the solution of an equation.

## What is an equation?

An equation is a statement that two expressions are equal. For example, the expression $5+3$ is equal to the expression $6+2$ (because they both equal $8$), so we can write the following equation:
$5+3=6+2$
All equations have an equal sign ($=$). The $=$ sign is not an operator like addition ($+$) or subtraction ($-$) symbols. The equal sign doesn't tell us what to do. It only tells us that two expressions are equal. For example, in:
$6-2=3+1$
The $-$ sign tells us what to do with $6$ and $2$: subtract $2$ from $6$. However, the $=$ sign does not tell us what to do with $6-2$ and $3+1$. It only tells us that they are equal.
Let's make sure we know the difference between an expression and an equation.
Which of these is an equation?

## True equations

All of the equations we just looked at were true equations because the expression on the left-hand side was equal to the expression on the right-hand side. A false equation has an $=$, but the two expressions are not equal to each other. For example, the following is a false equation.
$2+2=6$
When we see an equation that's not true, we can use the not equal sign ($\ne$) to show that the two expressions are not equal:
$2+2\ne 6$
Let's make sure we understand what a true equation is.
Which of these are true equations?

## Solutions to algebraic equations

All of the equations that we've looked at so far have included only numbers, but most equations include a variable. For example, the equation $x+2=6$ has a variable in it. Whenever we have an equation like this with a variable, we call it an algebraic equation.
For an algebraic equation, our goal is usually to figure out what value of the variable will make a true equation.
For the equation $x+2=6$, notice how $x=4$ creates a true equation and $x=3$ creates a false equation.
True equationFalse equation
$\begin{array}{rl}x+2& =6\\ \\ 4+2& \stackrel{?}{=}6\\ \\ 6& =6\end{array}$$\begin{array}{rl}x+2& =6\\ \\ 3+2& \stackrel{?}{=}6\\ \\ 5& \ne 6\end{array}$
Notice how we use the symbol $\stackrel{?}{=}$ when we're not sure if we have a true equation or a false equation.
The value of the variable that makes a true equation is called a solution to the equation. Going back to our example, $x=4$ is a solution of $x+2=6$ because it makes the equation true.

## Let's try a few problems

Problem 1
Select the solution to the equation.
$3+g=10$