Linear equation

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A linear equation in algebra is an equation which is constructed by equating two linear functions. An example of a linear equation in two (real) variables x and y is:

3x + y − 5 = −7x + 4y + 3.

Since linear equations can only contain functions that are linear in the variables of interest (i.e., first-degree polynomials), terms such as x2, y1/3, and xy are not allowed.

Forms of a linear equation

Complicated linear equations, such as the one above, can be rewritten using the laws of elementary algebra into several simpler forms. In what follows, capital letters represent constants (unspecified but fixed numbers), while x and y are the variables.

  • General form:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ax + By + C = 0}
Here 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, 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, is −C/B, and the slope of the line is −A/B.
  • Standard form:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ax + By = D}
Here A and B are not both equal to zero. As above, usually A ≥ 0. The standard form can be converted to the general form by setting C = −D.
  • Intercept form:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{x}{E} + \frac{y}{F} = 1}
Here E and F must be nonzero. The graph of the equation has x-intercept E and y-intercept F. The intercept form can be converted to the standard form by setting A = 1/E, B = 1/F and D = 1.
  • Slope-intercept form:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = Mx + F}
M is the slope of the line and F is the y-intercept.
  • Point-slope form:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - K = M(x - H)}
The graph passes through the point (H,K) and has slope M.
  • Two-point form:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - K = \frac{Q - K}{P - H} (x - H)}
Here PH. The graph passes through the points (H,K) and (P,Q), and has slope M = (QK) / (PH).
  • Parametric form:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = Tt + U} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = Vt + W}
Two simultaneous equations in terms of a variable parameter t, with slope M = V / T, x intercept (VUWT) / V and y intercept (WTVU) / T.
This can also be related to the two-point form with T = PH, U = H, V = QK, and W = K:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = (P - H)t + H} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.
  • Special cases:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = F}
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 F. There is no x-intercept, unless F = 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = E}
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 E. The slope is undefined. There is no y-intercept, unless E = 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 = 0}
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 consider the 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.

Note that if 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. An example would be 3x + 2 = 3x − 5.

In addition, there may be more than two variables in the equation or several simultaneous equations. For more information see System of linear equations.

Connection with linear functions and operators

In all of the named forms above (assuming the graph is not a vertical line), the variable y is a function of x, and the graph of this function is the graph of the equation.

In general, if the linear equation is written in the form y = f(x), then f has the properties:

f(x + y) = f(x) + f(y)
f(ax) = af(x)

where a is any scalar. A function which satisfies these properties is called a linear function, or more generally a linear operator.

Because of the linear property above, the solutions of linear equations of this kind can in general be described as a superposition of other solutions of the same equation. This makes linear equations particularly easy to solve and reason about.

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.

See also

External links

ar:معادلة خطية ca:Equació lineal cs:Lineární rovnice de:Lineare Gleichung et:Lineaarvõrrand fr:Équation linéaire it:Equazione lineare he:משוואה לינארית nl:Lineaire vergelijking ja:線型方程式 pl:Równanie liniowe ru:Линейное уравнение vi:Phương trình tuyến tính zh:一次方程