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Difference between revisions of "Equation"

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An '''equation''' is a [[relation]] which states that two [[expression]]s are equal, identical, or otherwise the same. Equations are easily identifiable because they are composed of two expressions with an equals sign ('=') between them.   
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{{WotWAnnounce|week=April 15-22}}
 
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An '''equation''' is a [[relation]] which states that two [[expression]]s are equal, identical, or otherwise the same. Equations are easily identifiable because they are composed of two expressions with an equals sign ('=') between them.   
 
  
 
Equations are similar to [[congruence]]s (which relate geometric figures instead of numbers) and other relationships which fall into the category of [[equivalence relation]]s.
 
Equations are similar to [[congruence]]s (which relate geometric figures instead of numbers) and other relationships which fall into the category of [[equivalence relation]]s.
  
 +
A unique aspect to equations is the ability to modify an original equation by performing operations (such as [[addition]], [[subtraction]], [[multiplication]], [[division]], and [[exponential function|powers]]).
 +
 +
It's important to note the distinction between an equation and an ''[[identity]]''. An identity in terms of some [[variable]]s states that two expressions are equal for every value of those variables: for example,
 +
 +
<math>x^2 - y^2 = (x - y)(x + y)</math>
 +
 +
is an identity that is true regardless of the values of <math>x</math> and <math>y</math> (and indeed holds in a [[commutative]] [[ring]]). However,
 +
 +
<math>x^2 = 4</math>
 +
 +
is an equation that is true for some particular values of <math>x</math>. 
 +
 +
In other words, one can say that an identity is a [[tautology | tautological]] equation
 +
 +
==Solve 2 variable equations in less than 5 seconds!!!==
 +
Video Link: https://youtu.be/pSYT95hSH6M
 +
==Word Problem AMC 8 Algebra Video==
 +
https://youtu.be/rQUwNC0gqdg?t=611
 +
 +
==Linear equations==
 +
 +
A linear equation is the simplest form of equations (with one or more variables). It has the form <math>ax + by + cz + ... = n</math>, where <math>a, b, c, ..., n</math> are numbers and <math>x, y, z, ...</math> are the variables. In linear equations, the variables are never squared or cubed as in [[quadratics]] or [[cubics]]. The graph of a linear equation is always a straight line. Some examples of linear equations are the following:
 +
 +
<math>2x + 5 = 20</math>
 +
 +
<math>2x + 4b = 16</math>
 +
 +
<math>ax + by = cz</math>
 +
 +
'''How to solve a linear equation with one variable'''
 +
 +
Linear equations with one variable are of the form <math>ax = b</math>, where <math>a, b</math> are numbers and <math>x</math> is a variable. The common recipe for solving linear equations with one variable is the following:
 +
 +
Solve the equation <math>5x + 4 = 24</math>
 +
 +
1. <math>5x + 4 = 24</math> (We have to distinguish the terms. On the right side we put the variable terms and on the left side we put the numbers.)
 +
 +
2. <math>5x + 4 - 4 = 24 - 4</math> (In order to achieve it, we have to add or subtract in both hand sides.)
 +
 +
3. <math>5x = 20</math> (Now we have to convert the equation into the form <math>ax = b</math>. We shall proceed to divide both sides by <math>a=5</math>.)
 +
 +
4. <math>\dfrac {5x}{5} = \dfrac {20}{5}</math>
 +
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5. <math>x = 4</math>
 +
 +
Hence the solution to our equation is <math>\boxed{x = 4}</math>.
 +
 +
'''How to solve a linear equation with multiple variables'''
 +
 +
Solve for <math>x</math> and <math>y</math> in the two equations <math>y-4=5x</math> and <math>6x-y=-4</math>
 +
 +
1. Solve for y in terms of x
 +
<math>y-4=5x</math>
 +
 +
<math>y=5x+4</math>
 +
 +
2. Substitute the value into the other equation and solve
 +
 +
<math>6x-y=-4</math>
 +
 +
<math>6x-(5x+4)=-4</math>
 +
 +
<math>6x-5x-4=-4</math>
 +
 +
<math>x-4=-4</math>
 +
 +
<math>x=\boxed0</math>
 +
 +
3. Substitute the value of x back in
 +
 +
<math>6x-y=-4</math>
 +
 +
<math>6(0)-y=-4</math>
 +
 +
<math>0-y=-4</math>
 +
 +
<math>-y=-4</math>
 +
 +
<math>y=\boxed4</math>
 +
 +
==Linear equations==
 +
 +
A linear equation is the simplest form of equations (with one or more variables). It has the form <math>ax + by + cz + ... = n</math>, where <math>a, b, c, ..., n</math> are numbers and <math>x, y, z, ...</math> are the variables. In linear equations, the variables are never squared or cubed as in [[quadratics]] or [[cubics]]. The graph of a linear equation is always a straight line. Some examples of linear equations are the following:
 +
 +
<math>2x + 5 = 20</math>
 +
 +
<math>2x + 4b = 16</math>
 +
 +
<math>ax + by = cz</math>
 +
 +
'''How to solve a linear equation with one variable'''
 +
 +
Linear equations with one variable are of the form <math>ax = b</math>, where <math>a, b</math> are numbers and <math>x</math> is a variable. The common recipe for solving linear equations with one variable is the following:
 +
 +
Solve the equation <math>5x + 4 = 24</math>
 +
 +
1. <math>5x + 4 = 24</math> (We have to distinguish the terms. On the right side we put the variable terms and on the left side we put the numbers.)
 +
 +
2. <math>5x + 4 - 4 = 24 - 4</math> (In order to achieve it, we have to add or subtract in both hand sides.)
 +
 +
3. <math>5x = 20</math> (Now we have to convert the equation into the form <math>ax = b</math>. We shall proceed to divide both sides by <math>a=5</math>.)
 +
 +
4. <math>\dfrac {5x}{5} = \dfrac {20}{5}</math>
 +
 +
5. <math>x = 4</math>
 +
 +
Hence the solution to our equation is <math>\boxed{x = 4}</math>.
 +
 +
'''How to solve a linear equation with multiple variables'''
 +
 +
Solve for <math>x</math> and <math>y</math> in the two equations <math>y-4=5x</math> and <math>6x-y=-4</math>
 +
 +
1. Solve for y in terms of x
 +
<math>y-4=5x</math>
 +
 +
<math>y=5x+4</math>
 +
 +
2. Substitute the value into the other equation and solve
 +
 +
<math>6x-y=-4</math>
 +
 +
<math>6x-(5x+4)=-4</math>
 +
 +
<math>6x-5x-4=-4</math>
 +
 +
<math>x-4=-4</math>
 +
 +
<math>x=\boxed0</math>
 +
 +
3. Substitute the value of x back in
 +
 +
<math>6x-y=-4</math>
 +
 +
<math>6(0)-y=-4</math>
 +
 +
<math>0-y=-4</math>
 +
 +
<math>-y=-4</math>
 +
 +
<math>y=\boxed4</math>
 +
 +
==Quadratic equations==
 +
 +
A quadratic equation has a degree of two. This means that the highest exponent in the equation is exactly two.
  
 
==See also==
 
==See also==
 
* [[Inequality]]
 
* [[Inequality]]
 +
 +
 +
[[Category:Definition]]
 +
[[Category:Equations]]

Latest revision as of 11:57, 26 July 2023

An equation is a relation which states that two expressions are equal, identical, or otherwise the same. Equations are easily identifiable because they are composed of two expressions with an equals sign ('=') between them.

Equations are similar to congruences (which relate geometric figures instead of numbers) and other relationships which fall into the category of equivalence relations.

A unique aspect to equations is the ability to modify an original equation by performing operations (such as addition, subtraction, multiplication, division, and powers).

It's important to note the distinction between an equation and an identity. An identity in terms of some variables states that two expressions are equal for every value of those variables: for example,

$x^2 - y^2 = (x - y)(x + y)$

is an identity that is true regardless of the values of $x$ and $y$ (and indeed holds in a commutative ring). However,

$x^2 = 4$

is an equation that is true for some particular values of $x$.

In other words, one can say that an identity is a tautological equation

Solve 2 variable equations in less than 5 seconds!!!

Video Link: https://youtu.be/pSYT95hSH6M

Word Problem AMC 8 Algebra Video

https://youtu.be/rQUwNC0gqdg?t=611

Linear equations

A linear equation is the simplest form of equations (with one or more variables). It has the form $ax + by + cz + ... = n$, where $a, b, c, ..., n$ are numbers and $x, y, z, ...$ are the variables. In linear equations, the variables are never squared or cubed as in quadratics or cubics. The graph of a linear equation is always a straight line. Some examples of linear equations are the following:

$2x + 5 = 20$

$2x + 4b = 16$

$ax + by = cz$

How to solve a linear equation with one variable

Linear equations with one variable are of the form $ax = b$, where $a, b$ are numbers and $x$ is a variable. The common recipe for solving linear equations with one variable is the following:

Solve the equation $5x + 4 = 24$

1. $5x + 4 = 24$ (We have to distinguish the terms. On the right side we put the variable terms and on the left side we put the numbers.)

2. $5x + 4 - 4 = 24 - 4$ (In order to achieve it, we have to add or subtract in both hand sides.)

3. $5x = 20$ (Now we have to convert the equation into the form $ax = b$. We shall proceed to divide both sides by $a=5$.)

4. $\dfrac {5x}{5} = \dfrac {20}{5}$

5. $x = 4$

Hence the solution to our equation is $\boxed{x = 4}$.

How to solve a linear equation with multiple variables

Solve for $x$ and $y$ in the two equations $y-4=5x$ and $6x-y=-4$

1. Solve for y in terms of x $y-4=5x$

$y=5x+4$

2. Substitute the value into the other equation and solve

$6x-y=-4$

$6x-(5x+4)=-4$

$6x-5x-4=-4$

$x-4=-4$

$x=\boxed0$

3. Substitute the value of x back in

$6x-y=-4$

$6(0)-y=-4$

$0-y=-4$

$-y=-4$

$y=\boxed4$

Linear equations

A linear equation is the simplest form of equations (with one or more variables). It has the form $ax + by + cz + ... = n$, where $a, b, c, ..., n$ are numbers and $x, y, z, ...$ are the variables. In linear equations, the variables are never squared or cubed as in quadratics or cubics. The graph of a linear equation is always a straight line. Some examples of linear equations are the following:

$2x + 5 = 20$

$2x + 4b = 16$

$ax + by = cz$

How to solve a linear equation with one variable

Linear equations with one variable are of the form $ax = b$, where $a, b$ are numbers and $x$ is a variable. The common recipe for solving linear equations with one variable is the following:

Solve the equation $5x + 4 = 24$

1. $5x + 4 = 24$ (We have to distinguish the terms. On the right side we put the variable terms and on the left side we put the numbers.)

2. $5x + 4 - 4 = 24 - 4$ (In order to achieve it, we have to add or subtract in both hand sides.)

3. $5x = 20$ (Now we have to convert the equation into the form $ax = b$. We shall proceed to divide both sides by $a=5$.)

4. $\dfrac {5x}{5} = \dfrac {20}{5}$

5. $x = 4$

Hence the solution to our equation is $\boxed{x = 4}$.

How to solve a linear equation with multiple variables

Solve for $x$ and $y$ in the two equations $y-4=5x$ and $6x-y=-4$

1. Solve for y in terms of x $y-4=5x$

$y=5x+4$

2. Substitute the value into the other equation and solve

$6x-y=-4$

$6x-(5x+4)=-4$

$6x-5x-4=-4$

$x-4=-4$

$x=\boxed0$

3. Substitute the value of x back in

$6x-y=-4$

$6(0)-y=-4$

$0-y=-4$

$-y=-4$

$y=\boxed4$

Quadratic equations

A quadratic equation has a degree of two. This means that the highest exponent in the equation is exactly two.

See also

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