Algebraic equations basics · EduSite Academy

▶ Variables, expressions, & equations

In algebra, we use variables like x, y, and z to represent unknown values. Expressions are combinations of variables and numbers, while equations equate two expressions. Variables can take on different values depending on the context, and we can evaluate expressions by substituting values for the variables.

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▶ Testing solutions to equations

In this math lesson, we learn how to solve an equation with one variable. We test different values for x to see which one satisfies the equation. By substituting x with the given options, we find the correct solution that makes both sides of the equation equal. Practice makes perfect!

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📄 Intro to equations

##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.

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##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 ($\neq$) to show that the two expressions are not equal:

$2+2 \neq 6$

Let's make sure we understand what a true equation is.

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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 $\purpleD{x = 4}$ creates a true equation and $\maroonD{x = 3}$ creates a false equation.

True equation	False equation
$\begin{align} \purpleD x +2 &= 6 \\\purpleD{4} +2 &\stackrel{?}{=} 6\\6 &= 6 \end{align}$	$\begin{align} \maroonD x +2 &= 6\\\maroonD{3} +2 &\stackrel{?}{=} 6\\5 &\neq 6 \end{align}$

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, $x=\purpleD{4}$ is a solution of $x + 2 = 6$ because it makes the equation true.

##Let's try a few problems

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