The properties used to solve an equation are the properties of the relationship of equality, reflexivity, symmetry and transitivity and the properties of operations.
These properties are as true in arithmetic and algebra as they are in propositional language.
This can be summarized as follows: If the same operation is performed on both sides of an equality, then the equality is still true.
The main properties of an equality are the logical properties that make it anequivalence relationship:
the reflexive property: for all values of x, we always have x = x;
the symmetric property: if x = y, then y = x;
the transitive property: if x = y and y = z, then x = z.
The properties of operations (specific to each operation) are the following:
Additive properties:
If x = y, then x + z = y + z, for all values of z;
If x + u = x + v, then u = v;
Therefore, for all values of z: if x = y, then x − z = y − z.
These properties can be generalized to all the operations as follows:
If x = y, then x z = y z, for all values of z not equal to the absorbing element of the operation, if x u = x v, then u = v, for all values of x not equal to the absorbing element of the operation.
Therefore, these properties are valid for the operations of multiplication and division, since the preceding restriction eliminates the case of division by zero.
Example
The variable y in the equation \(4x + 2y = 6x + 10\), can be isolated by following these steps:
commutative and associative properties of addition
\(⇔\)
\(− 2x + 2y = 10\)
Simplify
\(⇔\)
\(−2x + 2y − 2x = 10 − 2x\)
Additive inverse
\(⇔\)
\((− 2x − 2x) + 2y = 10 − 2x\)
Commutative and associative properties of addition
\(⇔\)
\(2y = 10 − 2x\)
calculation
\(⇔\)
\(\dfrac{2y}{2} = \dfrac{(10 − 2x)}{2}\)
Division Property of Equality
\(⇔\)
\(y = \dfrac{10}{2} − \dfrac{2x}{2}\)
distributive property of division over addition
\(⇔\)
\(y = 5 − x\)
calculation
As an enthusiast deeply versed in the intricacies of solving equations, I bring to light a wealth of knowledge on the properties instrumental in this mathematical endeavor. My expertise spans not only the abstract realm of propositional language but also delves into the concrete domains of arithmetic and algebra. Let's dissect the components of the article, elucidating the concepts and properties embedded in the process of equation-solving.
The foundation of equation-solving lies in the properties of equality, which can be succinctly captured through the lens of reflexivity, symmetry, and transitivity:
Reflexive Property:
For all values of (x), the statement (x = x) holds true. This property establishes the innate equality of any quantity with itself.
Symmetric Property:
If (x = y), then (y = x). This property underlines the interchangeability of the sides of an equation without altering its validity.
Transitive Property:
If (x = y) and (y = z), then (x = z). The transitive property enables the chaining of equalities, allowing the derivation of relationships between different variables.
Complementing these logical properties are the properties of operations, which are specific to each operation:
Additive Properties:
If (x = y), then (x + z = y + z) for all values of (z).
If (x + u = x + v), then (u = v).
Similarly, for subtraction: if (x = y), then (x - z = y - z).
Generalized Properties:
These can be extended to multiplication and division with some considerations:
If (x = y), then (xz = yz) for all values of (z) not equal to the absorbing element of the operation.
If (xu = xv), then (u = v) for all values of (x) not equal to the absorbing element of the operation.
The absorbing element consideration is crucial, especially in division, where division by zero is undefined.
Now, let's apply these properties to an example equation: (4x + 2y = 6x + 10).
The solution involves a series of steps, invoking properties such as additive inverse, commutative and associative properties of addition, simplification, and the Division Property of Equality:
Additive Inverse: (4x + 2y - 6x = 6x + 10 - 6x)
Commutative and Associative Properties: ((-2x + 2y) = 10)
Simplify: (-2x + 2y - 2x = 10 - 2x)
Additive Inverse: ((-4x + 2y) = 10 - 2x)
Commutative and Associative Properties: (2y = 10 - 2x)
Calculation: (y = \frac{10}{2} - \frac{2x}{2})
Distributive Property: (y = 5 - x)
The meticulous application of these properties transforms the equation, isolating the variable (y) and revealing its value as (5 - x). This exemplifies the power and versatility of the properties of equality and operations in solving mathematical equations.
These properties include the associative property, commutative property, distributive property, identity property, inverse property, reflexive property, symmetric property, and transitive property. These properties or axioms are used to solve equations and manipulate equations in algebra.
If we end up with the same term on both sides of the equal sign, such as 4 = 4 or 4x = 4x, then we have infinite solutions. If we end up with different numbers on either side of the equal sign, as in 4 = 5, then we have no solutions.
The properties of inequality are: addition property - if a > b, then a + c > b + c. subtraction property - if a > b, then a - c > b - c. multiplication property - if a > b and c > 0, then ac > bc, or when c < 0, then ac < bc.
There are four number properties: commutative property, associative property, distributive property and identity property. Number properties are only associated with algebraic operations that are addition, subtraction, multiplication and division.
The properties of equality are required to solve problems in algebra and other problems in math. The properties of equality describe the relation between the two sides of an equation and state that the two sides remain equal even after applying the same arithmetic operation on each side.
An algebraic proof is a list of justifications for each step when solving for a variable in algebra. These justifications are the reasons why a certain step moves to the next step. The purpose of showing the algebraic reasoning behind the solution to a problem is to show that it was done accurately.
When solving an inequality: • you can add the same quantity to each side • you can subtract the same quantity from each side • you can multiply or divide each side by the same positive quantity If you multiply or divide each side by a negative quantity, the inequality symbol must be reversed. So the solution is x > −1.
The last type of equation is known as a contradiction, which is also known as a No Solution Equation. This type of equation is never true, no matter what we replace the variable with. As an example, consider 3x + 5 = 3x - 5. This equation has no solution.
In order to solve equations, you need to work out the value of the unknown variable by adding, subtracting, multiplying or dividing both sides of the equation by the same value. Combine like terms. Simplify the equation by using the opposite operation to both sides. Isolate the variable on one side of the equation.
We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.
Introduction: My name is Aracelis Kilback, I am a nice, gentle, agreeable, joyous, attractive, combative, gifted person who loves writing and wants to share my knowledge and understanding with you.
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