Linear equation

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Graph sample of linear equations in two variables.

In mathematics, a linear equation is an equation that may be put in the form

${\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}+c=0,}$

where ${\displaystyle x_{1},\ldots ,x_{n}}$ are the variables or unknowns, and ${\displaystyle c,a_{1},\ldots ,a_{n}}$ are coefficients, which are often real numbers, but may be parameters, or even any expression that does not contain the unknowns. In other words, a linear equation is obtained by equating to zero a linear polynomial.

The solutions of such an equation are the values that, when substituted to the unknowns, make the equality true.

The case of one unknown is of a particular importance, and it is frequent that linear equation refers implicitly to this particular case, that is to an equation that may be written in the form

${\displaystyle ax+b=0.}$

If a ≠ 0 this linear equation has the unique solution

${\displaystyle x=-{\frac {b}{a}}}$

The solutions of a linear equation in two variables form a line in the Euclidean plane, and every line may be defined as the solutions of a linear equation. This is the origin of the term linear for qualifying this type of equations. More generally, the solutions of a linear equation in n variables form a hyperplane (of dimension n – 1) in the Euclidean space of dimension n.

Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations.

This article considers the case of a single equation with real coefficients, for which one studies the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficient and solutions in any field. For the case of several simultaneous linear equations, see System of linear equations.

One variable

A linear equation in one unknown x may always be rewritten

${\displaystyle ax=b.}$

If a ≠ 0, there is a unique solution

${\displaystyle x={\frac {b}{a}}.}$

If a = 0, then, if b = 0, every number is a solution of the equation, and, if b ≠ 0, there are no solutions (and the equation is said to be inconsistent).

Two variables

A common linear equation in two variables x and y is the relation that links the argument and the value of a linear function:

${\displaystyle y=mx+y_{0},}$

where m and ${\displaystyle y_{0}}$ are real numbers. The graph of such a linear function is thus the set of the solutions of this linear equation, which is a line in the Euclidean plane of slope m and y-intercept ${\displaystyle y_{0}}$.

Every linear equation in x and y may be rewritten

${\displaystyle ax+by+c=0,}$

where a and b are not both zero. The set of the solutions form a line in the Euclidean plane, which is the graph of a linear function if and only if b ≠ 0.

Using the laws of elementary algebra, linear equations in two variables may be rewritten in several standard forms that are described below, which are often referred to as "equations of a line". In what follows, x, y, t, and θ are variables; other letters represent constants (fixed numbers).

General (or standard) form

In the general (or standard^[1]) form the linear equation is written as:

${\displaystyle Ax+By=C,\,}$

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. The general form is sometimes written as:

${\displaystyle ax+by+c=0,\,}$

where a and b are not both equal to zero. The two versions can be converted from one to the other by moving the constant term to the other side of the equal sign.

Slope–intercept form

${\displaystyle y=mx+b,}$

where m is the slope of the line and b is the y intercept, which is the y coordinate of the location where the 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 passes through the point (0, b) and extends to the left and right at a slope of m. Vertical lines, having undefined slope, cannot be represented by this form.

A corresponding form exists for the x intercept, though it is less-used, since y is conventionally a function of x:

${\displaystyle x=ny+a.}$

Analogously, horizontal lines cannot be represented in this form. If a line is neither horizontal nor vertical, it can be expressed in both these forms, with ${\displaystyle m\cdot n=1}$, so ${\displaystyle m=1/n}$. Expressing y as a function of x gives the form:

${\displaystyle y=m(x-a),}$

which is equivalent to the polynomial factorization of the y intercept form. This is useful when the x intercept is of more interest than the y intercept. Expanding both forms shows that ${\displaystyle b=-ma}$, so ${\displaystyle a=-b/m}$, expressing the x intercept in terms of the y intercept and slope, or conversely.

Point–slope form

${\displaystyle y-y_{1}=m(x-x_{1}),\,}$

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, y − y₁) is proportional to the difference in the x coordinate (that is, x − x₁). The proportionality constant is m (the slope of the line).

Two-point form

${\displaystyle y-y_{1}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}(x-x_{1}),\,}$

where (x₁, y₁) and (x₂, y₂) are two points on the line with x₂ ≠ x₁. This is equivalent to the point-slope form above, where the slope is explicitly given as (y₂ − y₁)/(x₂ − x₁).

Multiplying both sides of this equation by (x₂ − x₁) yields a form of the line generally referred to as the symmetric form:

${\displaystyle (x_{2}-x_{1})(y-y_{1})=(y_{2}-y_{1})(x-x_{1}).\,}$

Expanding the products and regrouping the terms leads to the general form:

${\displaystyle x\,(y_{2}-y_{1})-y\,(x_{2}-x_{1})=x_{1}y_{2}-x_{2}y_{1}}$

Using a determinant, one gets a determinant form, easy to remember:

${\displaystyle {\begin{vmatrix}x&y&1\\x_{1}&y_{1}&1\\x_{2}&y_{2}&1\end{vmatrix}}=0\,.}$

Intercept form

${\displaystyle {\frac {x}{a}}+{\frac {y}{b}}=1,\,}$

where a and b must be nonzero. The graph of the equation has x-intercept a and y-intercept b. The intercept form is in standard form with A/C = 1/a and B/C = 1/b. Lines that pass through the origin or which are horizontal or vertical violate the nonzero condition on a or b and cannot be represented in this form.

Matrix form

Using the order of the standard form

${\displaystyle Ax+By=C,\,}$

one can rewrite the equation in matrix form:

${\displaystyle {\begin{pmatrix}A&B\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}C\end{pmatrix}}.}$

Further, this representation extends to systems of linear equations.

${\displaystyle A_{1}x+B_{1}y=C_{1},\,}$

${\displaystyle A_{2}x+B_{2}y=C_{2},\,}$

becomes:

${\displaystyle {\begin{pmatrix}A_{1}&B_{1}\\A_{2}&B_{2}\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}C_{1}\\C_{2}\end{pmatrix}}.}$

Since this extends easily to higher dimensions, it is a common representation in linear algebra, and in computer programming. There are named methods for solving a system of linear equations, like Gauss-Jordan which can be expressed as matrix elementary row operations.

Parametric form

${\displaystyle x=Tt+U\,}$

and

${\displaystyle y=Vt+W.\,}$

These are 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:

${\displaystyle x=(p-h)t+h\,}$

and

${\displaystyle y=(q-k)t+k.\,}$

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.

2D vector determinant form

The equation of a line can also be written as the determinant of two vectors. If ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ are unique points on the line, then ${\displaystyle P}$ will also be a point on the line if the following is true:

${\displaystyle \det({\overrightarrow {P_{1}P}},{\overrightarrow {P_{1}P_{2}}})=0.}$

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 determinant equals zero then the parallelogram has no area, and that will happen when two vectors are on the same line.

To expand on this we can say that ${\displaystyle P_{1}=(x_{1},\,y_{1})}$, ${\displaystyle P_{2}=(x_{2},\,y_{2})}$ and ${\displaystyle P=(x,\,y)}$. Thus ${\displaystyle {\overrightarrow {P_{1}P}}=(x-x_{1},\,y-y_{1})}$ and ${\displaystyle {\overrightarrow {P_{1}P_{2}}}=(x_{2}-x_{1},\,y_{2}-y_{1})}$, then the above equation becomes:

${\displaystyle \det {\begin{pmatrix}x-x_{1}&y-y_{1}\\x_{2}-x_{1}&y_{2}-y_{1}\end{pmatrix}}=0.}$

Thus,

${\displaystyle (x-x_{1})(y_{2}-y_{1})-(y-y_{1})(x_{2}-x_{1})=0.}$

Ergo,

${\displaystyle (x-x_{1})(y_{2}-y_{1})=(y-y_{1})(x_{2}-x_{1}).}$

Then dividing both side by ${\displaystyle (x_{2}-x_{1})}$ would result in the “Two-point form” shown above, but leaving it here allows the equation to still be valid when ${\displaystyle x_{1}=x_{2}}$.

Special cases

${\displaystyle y=b\,}$

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.

${\displaystyle x=a\,}$

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 every real number is a y-intercept. This is the only type of straight line which is not the graph of a function (it obviously fails the vertical line test).

Connection with linear functions

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

• Additivity: ${\displaystyle f(x_{1}+x_{2})=f(x_{1})+f(x_{2})\ }$
• Homogeneity of degree 1: ${\displaystyle f(ax)=af(x),\,}$

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, when written in this manner, produce functions which will have neither property above and hence are not linear functions in this sense. They are known as affine functions.

Example

An everyday example of the use of different forms of linear equations is computation of tax with tax brackets. This is commonly done with a progressive tax computation, using either point–slope form or slope–intercept form.

More than two variables

A linear equation can involve more than two variables. Every linear equation in n unknowns may be rewritten

${\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}=b,}$

where, a₁, a₂, ..., a_n represent numbers, called the coefficients, x₁, x₂, ..., x_n are the unknowns, and b is called the constant term. When dealing with three or fewer variables, it is common to use x, y and z instead of x₁, x₂ and x₃.

If all the coefficients are zero, then either b ≠ 0 and the equation does not have any solution, or b = 0 and every set of values for the unknowns is a solution.

If at least one coefficient is nonzero, a permutation of the subscripts allows one to suppose a₁ ≠ 0, and rewrite the equation

${\displaystyle x_{1}={\frac {b}{a_{1}}}-{\frac {a_{2}}{a_{1}}}x_{2}-\cdots -{\frac {a_{n}}{a_{1}}}x_{n}.}$

In other words, if a_i ≠ 0, one may choose arbitrary values for all the unknowns except x_i, and express x_i in term of these values.

If n = 3 the set of the solutions is a plane in a three-dimensional space. More generally, the set of the solutions is an (n – 1)-dimensional hyperplane in a n-dimensional Euclidean space (or affine space if the coefficients are complex numbers or belong to any field).