Властивості додавання і віднімання тотожностей

$$\textbf{Addition and Subtraction Properties of Equality}$$

The Addition and Subtraction Properties of Equality are keys to manipulating equations.

They state that when adding or subtracting the same number from both sides of the equation, the equation still holds!

When is this technique useful? For example, if we are evaluating the following equation: $$x+4 = 10$$ We want to have only $x$ on the left, that way we find out what it is. We know if we subtract $4$ on both sides of the equation, the equation still holds. $$\begin{align} x+4\,\color{#8925AE}{-\,4} &= 10 \,\color{#8925AE}{-\,4}\\ x &= 6 \end{align}$$

Adding and Subtracting Scale of Equality

We can visualize any equation as a scale. Here is a demonstration of how that works. If we start with the equation $x-2=5$, the scale is balanced, and it would look like this:

When we solve for $x$, we must add $2$ ($2$ blue circles) to both sides of the equation. This will force the scale to remain balanced. We see that modeled below.

This gives us our rule. Whatever we do to one side we have to do to the other.

If we only added two to the left side, the scale would move out of balance, leaning towards the left. Look at the top scale in the diagram below to see this. If we only added two to the right side, the scale would move out of balance, leaning towards the right. Look at the bottom scale in the diagram below to understand this.

Let's look at a couple more examples of this using different numbers in the problems.

$$\begin{align} x+9&=15\\ x+9-{\color{#8925AE}9}&=15 -{\color{#8925AE}9}\\ x&=6 \end{align}$$ $$\begin{align} 45&=25+x\\ 45-\color{#29A603}{25}&=25 +x -\color{#29A603}{25}\\ 20&=x \end{align}$$

Try out a problem for yourself!

(Video) The Addition Property

by mathman1024

This video by mathman1024 explains the addition property.

$\large{\text{Summary}}$

The Addition Property states that: $$\begin{gathered} a=b\\ \text{is equivalent to}\\ a\,{\color{#8925AE}{+\,c}}=b\,{\color{#8925AE}{+\,c}}\\ \text{or}\\ a\,{\color{#8925AE}{ - \,c}}=b\,{\color{#8925AE}{-\,c}} \end{gathered}$$ Basically, whatever you do to one side of the equal sign, you have to do to the other side.

To see this in an example, he looks at the problem: $x+21=58$. $$\begin{align} x+21&=58\\ x+21\,{\color{#8925AE}{- \,21}}&=58\,{\color{#8925AE}{-\,21}}\\ x&=37 \end{align}$$ As you can see, we wanted to get the variable $x$ by itself so we subtracted $21$. But to keep the equation balanced, or true, we had to also subtract $21$ from the other side.

You also should follow these rules for solving quadratic equations!

Addition and Subtraction Properties of Equality

Everyone is familiar with the basic idea that $1+1=2$. This lesson delves deeper into the equals sign that separates $1+1$ and $2$.

The equals sign acts like a balance on a see-saw. Everything on the left of it must be the same weight as everything on the right, otherwise the see-saw is unbalanced and tips over!

In the $1+1=2$ example, the two ones on the Left Hand Side (LHS) have the same weight as the one two on the Right Hand Side (RHS).

We can use the equivalency to create a balance with variables too! For instance, $x+x=2x$, or $x-x=0$, or even $(3x+4)-(2x-3)=(x+7)$

Properties of Equality

The first property of equality is the Reflexive Property, which states that: $$a=a\text{ for all }a\text{'s}$$ This is sort of obvious, something is always equal to itself. If I wrote $2=2$ you would probably say "Duh."

Next, is the Symmetric Property, which states that: $$\text{if }a=b\text{ then }b=a$$ This is also a little obvious. If $1+1=2$, then it follows that $2=1+1$ as well.

We also have the Transitive Property, which states that: $$\text{if }a=b\text{ and }b=c\text{ then }a=c$$ Think about it in a real world context. Imagine Timmy has the same amount of money as Martha, and Martha has the same amount of money as Javier.

Now that we know the fundamental properties of equality, we can use the Addition and Subtraction properties on them as well!

Take this equation: $$x+3=6$$ How do we find out what $x$ equals? Simple. We add $-3$ to both sides of the equation: $$\begin{align} x+3-3&=6-3\\ x&=3 \end{align}$$ By adding $-3$ to both sides of the equation, the equality is maintained because we change both sides by the same amount! These are both equivalent equations!

Suppose instead we have $$3x+6 = 4x+5$$ We can use these properties to easily solve for $x$! $$\begin{align} 3x+6-3x&=4x+5-3x\\ 6 &= x+5\\ x+5 &= 6\\ x+5-5&=6-5\\ x &= 1 \end{align}$$

We'll follow these rules when solving equations and inequalities.