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Reference

Rule

Adding the same number to both sides of an equation results in an equivalent equation. Let $a,$ $b,$ and $c$ be real numbers.

If $a=b,$ then $a+c=b+c.$

$x−3=5 $

By adding $3$ to both sides of the equation, the variable $x$ can be isolated and the solution to the equation can be found.
The Addition Property of Equality also holds true if $a,$ $b,$ and $c$ are complex numbers.Rule

Subtracting the same number from both sides of an equation results in an equivalent equation. Let $a,$ $b,$ and $c$ be real numbers.

If $a=b,$ then $a−c=b−c.$

$x+2=7 $

By subtracting $2$ from both sides of the equation, the variable $x$ can be isolated and the solution to the equation can be found.
$x+2=7$

SubEqn

$LHS−2=RHS−2$

$x+2−2=7−2$

Simplify left-hand side

$x+2 −2 =7−2$

SubTerms

Subtract terms

$x=5$

Rule

Given an equation, multiplying each side of the equation by the same number yields an equivalent equation. Let $a,$ $b,$ and $c$ be real numbers.

If $a=b,$ then $a×c=b×c.$

$x÷4x÷4×4x =2=2×4=8 $

Here, by multiplying both sides of the equation by $4,$ the variable $x$ was isolated and the solution of the equation was found. Note that the Multiplication Property of Equality also holds true if $a,$ $b,$ and $c$ are complex numbers.Rule

Dividing each side of an equation by the same nonzero number yields an equivalent equation. Let $a,$ $b,$ and $c$ be real numbers.

If $a=b$ and $c =0,$ then $a÷c=b÷c.$

$5x5x÷5x =10=10÷5=2 $

As can be observed, by dividing both sides of the equation by $5,$ the variable $x$ was isolated and the solution of the equation was found. Note that the Division Property of Equality also holds true if $a,$ $b,$ and $c$ are complex numbers.Rule

For any real number, the number is equal to itself.

$a=a$

$x+3=8 $

The sum on the left-hand side can be interpreted as a number, so this sum has to equal $8.$ This implies that $x$ must be equal to $5.$Rule

For all real numbers, the order of an equality does not matter. Let $a$ and $b$ be real numbers.

If $a=b,$ then $b=a.$

This property is an axiom, so it does not need a proof to be accepted as true. The Symmetric Property of Equality also holds true if $a$ and $b$ are complex numbers.

Rule

For all real numbers, if two numbers are equal to the same number, then they are equal to each other. Let $a,$ $b,$ and $c$ be real numbers.

If $a=b$ and $b=c,$ then $a=c.$

${x=5y−305y−30=20 ⇒x =20 $

Since this property is an axiom, it does not need a proof to be accepted as true. The Transitive Property of Equality also holds true if $a,$ $b,$ and $c$ are complex numbers.Rule

If two real numbers are equal, then one can be substituted for another in any expression.

If $a=b,$ then $a$ can be substituted for $b$ in any expression.

${2x+1=20y(2x+1)=40 (I)(II) $

▼

Solve by substitution

${2x+1=20y=2 $

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