What is an equation?

 

What is an equation?

An equation is a statement that two expressions are equal. For example, the expression $5 + 3$5, plus, 3 is equal to the expression $6 + 2$6, plus, 2 (because they both equal $8$8), so we can write the following equation:

$5 + 3 = 6 + 2$5, plus, 3, equals, 6, plus, 2

All equations have an equal sign ($=$equals). The $=$equals sign is not an operator like addition ($+$plus) or subtraction ($-$minus) 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$6, minus, 2, equals, 3, plus, 1

The $-$minus sign tells us what to do with $6$6 and $2$2: subtract $2$2 from $6$6. However, the $=$equals sign does not tell us what to do with $6-2$6, minus, 2 and $3+1$3, plus, 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?

Choose 1 answer:

Choose 1 answer:

• (Choice A)

  A

  $4$4

  This is an expression.
• (Choice B, Checked, Correct)

  CORRECT (SELECTED)

  $9 - 5 = 4$9, minus, 5, equals, 4

  This is an equation. It has an $=$equals sign.
• (Choice C)

  C

  $9 - 5$9, minus, 5

  This is an expression.

Correct

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 $=$equals, but the two expressions are not equal to each other. For example, the following is a false equation.

$2+2=6$2, plus, 2, equals, 6

When we see an equation that's not true, we can use the not equal sign ($\neq$does not equal) to show that the two expressions are not equal:

$2+2 \neq 6$2, plus, 2, does not equal, 6

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

Which of these are true equations?

Choose all answers that apply:

Choose all answers that apply:

• (Choice A)

  A

  $7 - 2 = 7 + 1$7, minus, 2, equals, 7, plus, 1

  $\begin{aligned} 7-2 &\stackrel{?}=7+1 \\\\ 5 &\neq 8 \end{aligned}$
• (Choice B, Checked, Correct)

  CORRECT (SELECTED)

  $10 = 8 + 2$10, equals, 8, plus, 2

  $\begin{aligned} 10 &\stackrel{?}=8+2 \\\\ 10 &\stackrel{\checkmark}=10 \end{aligned}$
• (Choice C, Checked, Correct)

  CORRECT (SELECTED)

  $8 + 2 = 5 + 5$8, plus, 2, equals, 5, plus, 5

  $\begin{aligned} 8+2 &\stackrel{?}=5+5 \\\\ 10 &\stackrel{\checkmark}=10 \end{aligned}$

Correct

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 $x + 2 = 6$x, plus, 2, equals, 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 $x + 2 = 6$x, plus, 2, equals, 6, notice how $\purpleD{x = 4}$start color #7854ab, x, equals, 4, end color #7854ab creates a true equation and $\maroonD{x = 3}$start color #ca337c, x, equals, 3, end color #ca337c creates a false equation.

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

Notice how we use the symbol $\stackrel{?}{=}$equals, start superscript, question mark, end superscript 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, $x=\purpleD{4}$x, equals, start color #7854ab, 4, end color #7854ab is a solution of $x + 2 = 6$x, plus, 2, equals, 6 because it makes the equation true.