Standard Form Of a Line

Standard Form of a Line

The standard form for linear equations in two variables is Ax + By = C. For example, 2x + 3y = 5 is a linear equation in standard form. When an equation is given in this form, it's pretty easy to find both intercepts (x and y). This form is also very useful when solving systems of two linear equations.

The standard form of a line is simply a special way of writing the equation of a line. You are probably already familiar with the slope-intercept form of a line, y = mx + b. The standard form is just another way to write this equation, and is defined as Ax + By = C, where A, B, and C are real numbers, and A and B are both not zero. As you will see in the lesson below, every line can be expressed in this form. A more complicated case is when either the slope, the constant, or maybe even both are fractions. Since the coefficients on x and y are preferably written as integers, this means that you have to “clear fractions”. 

The standard form for linear equations in two variables is Ax + By = C. For example, 2x + 3y = 5 is a linear equation in standard form. When an equation is given in this form, it's pretty easy to find both intercepts (x and y). This form is also very useful when solving systems of two linear equations.

Place Your Order

Finding the standard form of a line

Two lines, line segments, or rays (or any combination of those) are parallel if they never meet and are always the same distance apart. Both lines have to be in the same plane (be coplanar).

You encounter parallel lines in geometry, of course, but also in everyday life. The lines of notebook paper are parallel. Sides of doors, edges of cereal boxes, and the floorboards of a home are parallel.

Finding the slope from the standard form of a line

Slope is the rate of change of one variable with respect to another variable. For instance, when you're driving your car, taking a walk, or riding your bike, you're traveling at a certain speed. That speed is the rate of change of your distance with respect to time. You probably never thought of your speed as a slope before, but that's exactly what it is. If you have an equation representing your distance in terms of time, the slope associated with that equation is your speed.

In mathematical terms, the slope is the rate of change of y with respect to the xslope-intercept form, . When dealing with linear equations, we can easily identify the slope of the line represented by the equation by putting the equation in y = mx + b, where m is the slope and b is the y-intercept.

Thus, if we have a linear equation in slope-intercept form, it's easy to see that the slope of the line is the constant in front of the x variable. For example, suppose that price of a product over x years is modeled by the equation y = 0.2x + 5, where y is the price of the product, and x is the number of years that have passed since the company started selling that product.

Get Assistance

Deriving the Standard Form for a Linear Equation

The standard form for linear equations in two variables is Ax+By =C. For example, 2x+3y=5 is a linear equation in standard form. When an equation is given in this form, it's pretty easy to find both intercepts (x and y). This form is also very useful when solving systems of two linear equations.

The standard form of a line can be particularly helpful when solving a system of equations. For instance, when using the elimination method to solve a system of equations, we can easily align the variables using standard form.

In the case of two variables, each solution may be interpreted as the Cartesian coordinates of a point of the Euclidean plane. The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term linear for describing this type of equations. More generally, the solutions of a linear equation in n variables form a hyperplane (a subspace 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 coefficients from the field of real numbers, for which one studies the real solutions. All of its content applies to complex solutions and, more generally, for linear equations with coefficients and solutions in any field.

Proposition 1.27 of Euclid's Elements, a theorem of absolute geometry, proves that if the angles of a pair of alternate angles of a transversal are congruent then the two lines are parallel (non-intersecting).

The functions whose graph is a line are generally called linear functions in the context of calculus. However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. So, for this definition, the above function is linear only when c = 0, that is when the line passes through the origin. For avoiding confusion, the functions whose graph is an arbitrary line are often called affine functions.

With this interpretation, all solutions of the equation form a line, provided that a as well as b are not both zero. Conversely, every line is the set of all solutions of a linear equation.

The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line.