Graph sample of linear equations.

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.

Linear equations can have one or more variables. Linear equations occur with great regularity in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state. Linear equations do not include exponents.

Linear equations in two variables[link]

A common form of a linear equation in the two variables x and y is

Failed to parse (Missing texvc executable; please see math/README to configure.): y = mx + b,\,

where m and b designate constants. The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In this particular equation, the constant m determines the slope or gradient of that line, and the constant term "b" determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.

Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than 1) or other function of a variable, equations involving terms such as xy, x², y^1/3, and sin(x) are nonlinear.

Forms for 2D linear equations[link]

Linear equations can be rewritten using the laws of elementary algebra into several different forms. These equations are often referred to as the "equations of the straight line." In what follows, x, y, t, and θ are variables; other letters represent constants (fixed numbers).

General form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): Ax + By + C = 0, \,

where A and B are not both equal to zero. The equation is usually written so that A ≥ 0, by convention. The graph of the equation is a straight line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, the x-coordinate of the point where the graph crosses the x-axis (where, y is zero), is −C/A. If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where x is zero), is −C/B, and the slope of the line is −A/B.

Standard form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): Ax + By = C,\,

where A and B are not both equal to zero, A, B, and C are coprime integers, and A is nonnegative (if zero, B must be positive). The standard form can be converted to the general form, but not always to all the other forms if A or B is zero. It is worth noting that, while the term occurs frequently in school-level US algebra textbooks, most lines cannot be described by such equations. For instance, the line x + y = √2 cannot be described by a linear equation with integer coefficients since √2 is irrational.

Slope–intercept form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): y = mx + b,\,

where m is the slope of the line and b is the y-intercept, which is the y-coordinate of the location where line crosses the y axis. This can be seen by letting x = 0, which immediately gives y = b. It may be helpful to think about this in terms of y = b + mx; where the line originates at (0, b) and extends outward at a slope of m. Vertical lines, having undefined slope, cannot be represented by this form.

Point–slope form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): y - y_1 = m( x - x_1 ),\,

where m is the slope of the line and (x₁,y₁) is any point on the line.

The point-slope form expresses the fact that the difference in the y coordinate between two points on a line (that is, Failed to parse (Missing texvc executable; please see math/README to configure.): y - y_1

) is proportional to the difference in the x coordinate (that is, Failed to parse (Missing texvc executable; please see math/README to configure.): x - x_1 ). The proportionality constant is m (the slope of the line).

Two-point form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1),\,

where Failed to parse (Missing texvc executable; please see math/README to configure.): (x_1,y_1)

and Failed to parse (Missing texvc executable; please see math/README to configure.): (x_2,y_2)
are two points on the line with Failed to parse (Missing texvc executable; please see math/README to configure.): x_2
≠ Failed to parse (Missing texvc executable; please see math/README to configure.): x_1

. This is equivalent to the point-slope form above, where the slope is explicitly given as Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{y_2 - y_1}{x_2 - x_1} .

Intercept form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{x}{a} + \frac{y}{b} = 1,\,

where a and b must be nonzero. The graph of the equation has x-intercept a and y-intercept b. The intercept form can be converted to the standard form by setting A = 1/a, B = 1/b and C = 1.

Parametric form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): x = T t + U\,

and

Failed to parse (Missing texvc executable; please see math/README to configure.): y = V t + W.\,

Two simultaneous equations in terms of a variable parameter t, with slope m = V / T, x-intercept (VU−WT) / V and y-intercept (WT−VU) / T.

This can also be related to the two-point form, where T = p−h, U = h, V = q−k, and W = k:

Failed to parse (Missing texvc executable; please see math/README to configure.): x = (p - h) t + h\,

and

Failed to parse (Missing texvc executable; please see math/README to configure.): y = (q - k)t + k.\,

In this case t varies from 0 at point (h,k) to 1 at point (p,q), with values of t between 0 and 1 providing interpolation and other values of t providing extrapolation.

Polar form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): r=\frac{mr\cos\theta+b}{\sin\theta},

where m is the slope of the line and b is the y-intercept. When θ = 0 the graph will be undefined. The equation can be rewritten to eliminate discontinuities:

Failed to parse (Missing texvc executable; please see math/README to configure.): r\sin\theta=mr\cos\theta+b.\,

Normal form[link]

The normal for a given line is defined to be the shortest segment between the line and the origin. The normal form of the equation of a straight line is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): y \sin \theta + x \cos \theta - p = 0,\,

where θ is the angle of inclination of the normal, and p is the length of the normal. The normal form can be derived from general form by dividing all of the coefficients by

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{|C|}{-C}\sqrt{A^2 + B^2}.

This form is also called the Hesse standard form, after the German mathematician Ludwig Otto Hesse.

2D vector determinant form[link]

The equation of a line can also be written as the determinant of two vectors. If Failed to parse (Missing texvc executable; please see math/README to configure.): P_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): P_2
are unique points on the line, then Failed to parse (Missing texvc executable; please see math/README to configure.): P
will also be a point on the line if the following is true:

Failed to parse (Missing texvc executable; please see math/README to configure.): \det( \overrightarrow{P_1 P} , \overrightarrow{P_1 P_2} ) = 0.

One way to understand this formula is to use the fact that the determinant of two vectors on the plane will give the area of the parallelogram they form. Therefore, if the determinate equals zero then the parallelogram has no area, and that will happen when to vectors are on the same line.

To expand on this we can say that Failed to parse (Missing texvc executable; please see math/README to configure.): P_1 = (x_1 ,\, y_1) , Failed to parse (Missing texvc executable; please see math/README to configure.): P_2 = (x_2 ,\, y_2)

and Failed to parse (Missing texvc executable; please see math/README to configure.): P = (x ,\, y)

. Thus Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{P_1 P} = (x-x_1 ,\, y-y_1)

and Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{P_1 P_2} = (x_2-x_1 ,\, y_2-y_1)

, then the above equation becomes:

Failed to parse (Missing texvc executable; please see math/README to configure.): \det \begin{pmatrix}x-x_1&y-y_1\\x_2-x_1&y_2-y_1\end{pmatrix} = 0.

Thus,

Failed to parse (Missing texvc executable; please see math/README to configure.): ( x - x_1 )( y_2 - y_1 ) - ( y - y_1 )( x_2 - x_1 )=0.

Ergo,

Failed to parse (Missing texvc executable; please see math/README to configure.): ( x - x_1 )( y_2 - y_1 ) = ( y - y_1 )( x_2 - x_1 ).

Then dividing both side by Failed to parse (Missing texvc executable; please see math/README to configure.): ( x_2 - x_1 )

would result in the “Two-point form” shown above, but leaving it here allows the equation to still be valid when Failed to parse (Missing texvc executable; please see math/README to configure.): x_1 = x_2

.

Special cases[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): y = b\,

Horizontal Line y = b

This is a special case of the standard form where A = 0 and B = 1, or of the slope-intercept form where the slope M = 0. The graph is a horizontal line with y-intercept equal to b. There is no x-intercept, unless b = 0, in which case the graph of the line is the x-axis, and so every real number is an x-intercept.

Failed to parse (Missing texvc executable; please see math/README to configure.): x = a\,

Vertical Line x = a

This is a special case of the standard form where A = 1 and B = 0. The graph is a vertical line with x-intercept equal to a. The slope is undefined. There is no y-intercept, unless a = 0, in which case the graph of the line is the y-axis, and so every real number is a y-intercept.

Failed to parse (Missing texvc executable; please see math/README to configure.): y = y \

and Failed to parse (Missing texvc executable; please see math/README to configure.):  x = x\,

In this case all variables and constants have canceled out, leaving a trivially true statement. The original equation, therefore, would be called an identity and one would not normally consider its graph (it would be the entire xy-plane). An example is 2x + 4y = 2(x + 2y). The two expressions on either side of the equal sign are always equal, no matter what values are used for x and y.

Failed to parse (Missing texvc executable; please see math/README to configure.): e = f\,

In situations where algebraic manipulation leads to a statement such as 1 = 0, then the original equation is called inconsistent, meaning it is untrue for any values of x and y (i.e., its graph would be the empty set). An example would be 3x + 2 = 3x − 5.

Connection with linear functions[link]

A linear equation, written in the form y = f(x) whose graph crosses through the origin, that is whose y-intercept is 0, has the following properties:

Failed to parse (Missing texvc executable; please see math/README to configure.): f ( x_1 + x_2 ) = f ( x_1) + f ( x_2 )\

and

Failed to parse (Missing texvc executable; please see math/README to configure.): f ( a x ) = a f ( x ),\,

where a is any scalar. A function which satisfies these properties is called a linear function (or linear operator, or more generally a linear map). However, linear equations that have non-zero y-intercepts will have neither property above and hence are not linear functions in this sense.

Linear equations in more than two variables[link]

A linear equation can involve more than two variables. The general linear equation in n variables is:

Failed to parse (Missing texvc executable; please see math/README to configure.): a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b.

In this form, a₁, a₂, …, a_n are the coefficients, x₁, x₂, …, x_n are the variables, and b is the constant. When dealing with three or fewer variables, it is common to replace x₁ with just x, x₂ with y, and x₃ with z, as appropriate.

Such an equation will represent an (n–1)-dimensional hyperplane in n-dimensional Euclidean space (for example, a plane in 3-space).

In vector notation, this can be expressed as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{n} \cdot \overrightarrow{x} = \overrightarrow{n} \cdot \overrightarrow{x_0}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{n}

is a vector normal to the plane, Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{x}
are the coordinates of any point on the plane, and Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{x_0}
are the coordinates of the origin of the plane.

See also[link]

• Line (geometry)
• Quadratic equation
• Cubic equation
• Quartic equation
• Quintic equation
• Linear inequality
• Linear belief function

References[link]

External links[link]

• Algebraic Equations at EqWorld: The World of Mathematical Equations.
• [1] Video tutorial on solving one step to multistep equations
• Linear Equations and Inequalities Open Elementary Algebra textbook chapter on linear equations and inequalities.

This biographical article needs additional citations for verification. Please help by adding reliable sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately, especially if potentially libelous or harmful. (May 2010)

The Basics
The Basics at the Oxford Art Factory, 2010
Background information
Origin	Melbourne, Australia
Genres	Rock
Years active	2002–present
Labels	Independent, Albert Records
Associated acts	Gotye, Down Hills Home, Dog With Wheels, Blood Red Bird
Website	The Basics Official Website
Members
Wally De Backer Kris Schroeder Tim Heath
Past members
Michael Hubbard
Notable instruments
Lead guitar, drums, bass guitar.

The Basics ( /ˈbæzɪtʃ/ BAZZ-ich) are a three-piece band from Melbourne, Australia, formed by Wally De Backer and Kris Schroeder in 2002. Their style has been described as anything from 'indie-pop' to 'rock'n'roll' to 'pop-rock', though their records show they span a wide range of genres, including reggae, ska, country, and electro-pop. They are "recognised as one of Australia's hardest-working bands".^[1]

With close to 1000 shows in their ten-year career, their live performances are well-known for their sense of humour and energy.^[2]

Formation[link]

The Basics were formed after Kris met Wally at a party in Melbourne.^[3] The event was to see off the young aspiring producer, who was about to depart for Los Angeles, and together with a number of musician friends, the two jammed on some blues-rock standards. Later that night, they bonded over a mutual love of The Beatles, 70s and 80s cartoon theme songs and old Sierra adventure games.

Initially, they played around Melbourne as an acoustic guitar/drums combo, starting at The Opposition in Frankston and the House of Fools in Footscray, where in late 2002 they met Michael Hubbard and later invited him to join them on electric guitar.^[4] Though lacking any real experience with the instrument, Kris willingly purchased his first bass guitar, and the trio was born. During this time, the group start performing songs with more complicated three-part harmony.

During late 2002 and early 2003, The Basics recorded and released an album called Get Back through MGM Distribution. They began what was to become regular tours of the East and West coasts of Australia.

2004-2005: Current lineup[link]

Tim Heath joined the group in June 2004, after Michael decided to pursue his solo career, which has included being one half of Down Hills Home.^[4] Tim was reportedly the only person to reply to an advertisement in Melbourne's Beat Magazine looking for a new guitarist. Tim apparently answered the advertisement - which said the applicant "must have a love of 50s and 60s rock'n'roll" - thinking that he would be meeting "a couple of old guys to play with on the weekends over some beers". They soon recorded For Girls Like You.

The Basics in 2005 continued to tour and play small festivals, and after releasing For Girls Like You through MGM Distribution, they departed for a national tour of Australia, which lasted for two months and took them to every Australian state and territory. Stuart Padbury, a young Melbourne sound engineer, joined them on much of this tour. On returning, the band began demonstrating songs for a new album.

2006-2009: Touring and soundtracks[link]

In 2006, The Basics began recording what was to become known as Stand Out/Fit In. Taking several months, it was not completed until the end of the year, despite intentions to have it released in June. Instead, they held a successful 'album preview', where each of the tracks was demonstrated in 'rough form' accompanied by a music video for each (produced by Melbourne photographic artist James Bryans). The band also talked about the origins of the songs. This event brought a lot of attention to the band for the idea's originality. They pre-released a radio single 'Just Hold On' from the album. The band were invited to play the hugely popular Homebake Festival in Sydney, where they received a great response, after being introduced onto the stage by a short stocky Asian guy with big calves ("one of the highlights" wrote The Daily Telegraph).

In 2007, the band departed on a tri-state residency which saw them play Melbourne, Sydney and Brisbane every week for the month of April - a bold move which brought further attention to the band for its original idea.^[5] Riding on the wave of this tour, Schroeder, De Backer and Heath departed on their first overseas tour which saw them performing 26 shows around Japan, Norway and the United Kingdom.^[6] They also toured twice more up the East Coast, one of which was with popular Japanese band The Bawdies. Their song 'Rattle My Chain' was used in a series of commercials for Volvo Australia. The song 'Hey There!' featured in the British film The Bank Job - starring Jason Statham - and was used as the backing for a February 2007 short film directed by Tim Longhurst called The Rip. Their song 'Better' was featured in a surfing documentary about the invention of the shortboard in 1967.

In 2008, the band appeared on Australia's Channel 9 for Australia Day celebrations, performing two of their songs - 'Just Hold On' and 'Hey There!' - to an estimated audience of 2 million. Securing an Australian Government grant, they embarked on a two-month tour of Australia, specifically targeting "culturally underprivileged groups" in rural and regional schoolchildren and Indigenous communities. The programme had them holding masterclasses and performing alongside Indigenous groups in the remote Northern Territory and Queensland.^[7][8] Their initiative also had them raising money for the charity Lifeline^{[disambiguation needed ]}, their contribution to which was recognised with a plaque at the end of the tour. Season two of Californication starring David Duchovny featured their cover of the classic 'Have Love, Will Travel', most famously covered by 'The Sonics'.

In 2009, The Basics for the first time enjoyed the support of Australian radio network Triple J, which had reportedly snubbed the band's previous material.^[9] The singles 'With This Ship' and 'Like A Brother' were added to the station's playlist, and momentum carried The Basics overseas for a second tour of the United Kingdom and Norway, and also a series of shows in Dublin, Ireland. They were joined by the longtime front-of-house engineer Stuart Padbury, who had been mixing them since 2005. The band was often seen in the Grafton Street Mall in Dublin, busking to promote their concerts there. Their second show in Oslo, Norway, saw the band perform with Hot Tub, one of the members of the Norwegian boy band "Boyzvoice", in his first performance for several years. Season Eight of Scrubs featured the song 'Lookin' Over My Shoulder'.

Despite all this promise, the band's album Keep Your Friends Close - produced and part-recorded by Peter Cobbin at London's Abbey Road Studios - though enjoying many favourable reviews,^{[10][11][12][13]} failed to impress Richard Kingsmill, music director of Triple J. This effectively ended the band's apparent upward spiral and the resulting album launch tour was reportedly "trying".

2010: Hiatus and live album[link]

In 2010, following the band's fourth stint at the Tamworth Country Music Festival, and the recording of a yet-unreleased live album, an indefinite hiatus was announced on the band's websites. This was in part due to the impending release of a new album produced by Wally for his Gotye project, and also to reported tensions within the band following the previous year's disappointments.

However, this hiatus was soon broken with a two-month residency at the Northcote Social Club in Melbourne, which saw an apparent return to form.^[14] The band's EP "Wait For You" enjoys airplay on Triple J.^[15] The Basics also played a one-off show at Sydney's Oxford Art Factory.^[16]

On 20 August 2010, The Basics released their fourth LP /ðəˈbæzɪtʃ/, available free and exclusively from their website for digital download^[17]. The band had reportedly moved to expand their listenership through this free giveaway, though 300 "Deluxe" CD/vinyl versions have been printed. /ðəˈbæzɪtʃ/ was recorded at the Northcote Social Club on 6 February, "in front of a live studio audience", and like the previous record, was mixed by Peter Cobbin at Abbey Road Studios. Technically a self-titled album, the phonetics spell out the accurate pronunciation of the band’s name. Kris Schroeder: "For years we’ve worked at bringing our live energy into the studio and thus far it’s been a bit hit-and-miss. Our trick this time was to pull a switcheroo and bring the studio to us instead; the result is the best and closest to us we’ve sounded yet."

Melbourne thespian and Dog With Wheels bassist David Bramble - friends with Heath from university days - has occasionally joined the band on tour, playing keys, often while sporting a large moustache. Other guests have included Jake Mason (saxophone), also of Cookin' on 3 Burners and The Bamboos^{[disambiguation needed ]}, Gideon Brazil (saxophone, flute), Simon Imrei (guitar and vocals) and Monty MacKenzie (saxophone).

Covers have often made an appearance from various sources: to date, some of the bands covered have been Cream, The Kinks, The Rolling Stones, The Police, Neil Young, Bob Dylan, New Kids on the Block, Richard Berry, Sonny Curtis and The Crickets, JJ Cale, Harry Nilsson, Bill Scott, Ryan Adams and The Coasters (via The Beatles for their cover of Three Cool Cats).

Members[link]

Current

• Wally De Backer - drums, vocals (2002-present)
• Kris Schroeder - bass, vocals (2002-present)
• Tim Heath - guitar, vocals (2004-present)

Former

• Michael Hubbard - guitar, vocals (2002-2004)

Discography[link]

Albums[link]

• Get Back (15 January 2004)
• Stand Out/Fit In (5 May 2007)
• Keep Your Friends Close (25 September 2009)
• /ðəˈbæzɪtʃ/ (20 August 2010)

EP[link]

• For Girls Like You (11 July 2005)
• Lookin' Over My Shoulder (10 February 2007)
• Like A Brother (5 June 2009)
• Wait For You (21 May 2010)

Singles[link]

• Call It Rhythm & Blues (20 August 2004)
• Just Hold On
• With This Ship (22 February 2009)

Upcoming projects[link]

The band is taking a three year hiatus, with an announced return to the stage together in early 2014. Wally De Backer launched Making Mirrors under his Gotye act in August 2011, Tim Heath will be touring and recording with Blood Red Bird, and Kris Schroeder recorded a solo EP titled, Patience in the Face of Control, before leaving Australia to work with the Red Cross in Kenya for a two year sabbatical.

References[link]

External links[link]

• Official Website
• Myspace Website