2.02 Properties of equality
Manipulate equations by maintaining equality
We want to keep equations balanced so that the two sides of the equals sign remain equivalent. If we don't we could change what the equation means. Think of a balanced set of scales. The scale remains level when the weights on both sides of the scales are even. The same thing happens with equations.
This shows a balanced equation.
If we add a weight to one side and not to the other, then the scales will no longer be balanced.
Practice questions
Question 1
Beginning with the equation $x=14$x=14, write the new equation produced by subtracting $7$7 from both sides.
Make sure to simplify your answer, if possible.
Question 2
Beginning with the equation $3x=-8$3x=−8, write the new equation produced by multiplying both sides by $12$12.
The properties of equality for real numbers
What exactly are the properties of equality? Put simply, they're assumptions of how we can manipulate equations while maintaining truth. They're the foundation for most of mathematics.
The table below summarizes the properties and gives an example of how we might apply them in the context of algebra.
| Property of equality | Meaning | Example |
|---|---|---|
| Reflexive property | Anything is equal to itself. | $x=x$x=x |
| Symmetric property | The reverse equation is also true. | If $x=13$x=13 then $13=x$13=x. |
| Transitive property | Equality is transferable. | If $x=y$x=y and $y=z$y=z, then $x=z$x=z. |
| Addition property | Equals added to equals are still equal. | If $x=20$x=20 then $x+5=20+5$x+5=20+5. |
| Subtraction property | Equals subtracted from equals are still equal. | If $x=20$x=20 then $x-5=20-5$x−5=20−5. |
| Multiplication property | Equals multiplied by equals are still equal. | If $x=20$x=20 then $3x=3(20)$3x=3(20). |
| Division property | Equals divided by equals are still equal, as long as the division isn't by 0. | If $x=20$x=20 then $\frac{x}{5}=\frac{20}{5}$x5=205. |
Since these properties are true for any real number, they're also true for real number measurements, such as segment lengths, or angle measures.
Practice questions
Question 3
By which property are $9x=-27$9x=−27 and $\frac{9x}{9}=\frac{-27}{9}$9x9=−279 equivalent equations?
Multiplication property of equality.
AAddition property of equality.
B
Question 4
Fill in the blank so that the resulting statement is true.
The addition property of equality states that if $a=b$a=b then $a+c$a+c = $\editable{}$.