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Course: 6th grade (Illustrative Mathematics)>Unit 6

Lesson 2: Lesson 2: Truth and equations

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.
$3+g=10$

Want to join the conversation?

• what is a true equation and a false equation?
• I think it is what makes the equation true like for example - 6+B= 7 true equation would be 1 false because six plus one is seven, false equation would be 2 because six plus two would not equal seven it would equal eight.
• What do we do when its a decimal??
• The same thing you do with whole numbers
• is it a equation if its a fraction
• It can be, if it shows something like 1/2=2/4 (with an equal sign), but it is only an expression if it has no equal sign. For example, 3/8
• What is a true equation and a false??
• True is when the equation is correct. For example, if I were to write the equation 9+9 = 10+8, it would be true because both sides equal 18. However, if I were to write the equation 9+9 = 9+8, it would be false because one side equals 18 and one side equals 17.
• So it can still be an equation even if it has flopped Around?
• Yes; 2x + 3 = 4 = 4 = 3 + 2x, and I think you mean "flipped." :)
• what does the 3rd question in problem 3 mean
• It means 10=2w =?
Well, using common knowledge we know that 2 x ? = 10
If a variable is next to a number, has a floating period or has parenthesis we must multiply it.
HOPE IT HELPS!!
• how come we mostly use x? shouldn't it be something that makes sense like if its 5x6 shouldn't it be something like 5xS because the first letter of 6 is s?? not sure but just a thought.
• I am pretty sure it is because X is a relatively uncommon letter in the alphabet, for example if we used "a" more regularly it could get confusing.