Equation Of A Line Formula

Equation of a Line Formula

An equation of line represents a line in a coordinate system as a set of points. To form an equation of a line, the numerous points in the coordinate axis are represented by a set of variables x, y. It is possible to determine whether a given point lies on a line by using its equation. Line equations are linear equations of degree one. Students can discover how to find the Equation Of A Line Formula and the different forms of equation of a line. During their learning of the Mathematics curriculum, students will learn how to find the Equation Of A Line Formula and the different forms of the equation of a line.

What is the Equation of a Line?

The slope of a line and a point on the line can be used to form the Equation Of A Line Formula. One can learn more about the slope of the line and the needed point on the line to better understand how the Equation Of A Line Formula is formed. As an integer, fraction, or tangent of the angle the line makes with the positive x-axis, slope is the inclination of the line with the positive x-axis. In the coordinate system, a point has an x coordinate and a y coordinate. The coefficients of a linear equation are obtained by equating to zero a linear polynomial over some field. Solving such an equation involves finding values that, when substituted for unknowns, make the equality true.

There is only one solution when there is just one variable. This particular case, in which the unknown variable is called the variable, is often referred to as a linear equation implicitly.

For two variables, each solution can be interpreted as the Cartesian coordinates of a point on the Euclidean plane. A linear equation’s solutions form a line in the Euclidean plane, and every line is a set of all solutions to a linear equation in two variables. For this type of equation, the term linear is used. It has also been found that the solutions to a linear equation in n variables form a hyperplane (an n-dimensional subspace in n) in Euclidean space of n.

Mathematical equations and their applications in physics and engineering are often linear, partly because linear equations can approximate non-linear systems well.

This mathematical theme examines the real solutions of a single equation with coefficients from the field of real numbers. It covers complex solutions, as well as linear equations with coefficients and solutions in any field.

Standard Form of Equation of a Line

An Equation Of A Line Formula is written as ax + by + c = 0. A and b represent coefficients, x and y represent variables, and c represents a constant. A point on the line represented in the coordinate plane is defined by the values of x and y. To write this standard form of a line equation, students must follow these quick rules.

• First, the x term is written, then the y term, and finally the constant term.
• It is important to write the coefficients and constant values as integers, not fractions or decimals.
• Positive integers are always used to represent ‘a’, the coefficient of x.

The Equation Of A Line Formula is ax + by + c = 0 in standard form

where,

• Coefficients a and b
• Variables x and y
• Constant c

A line equation can be expressed in five different ways. This information is transformed and presented in a standard format.

Different Forms of Equation of a Line

Based on the parameters of a straight line, there are about five basic ways to write the Equation Of A Line Formula. The following are the different forms used to find and represent a line’s equation:

• Point Slope Form
• Intercept form
• Normal form
• Two Point Form
• Slope-intercept form

Explore each of these forms of the equation of a line in more detail. Along with the information about the Equation Of A Line Formula, Extramarks also provides students with various learning tools like-

• With Extramarks, students can practice and analyse all subjects with chapter-by-chapter worksheets, interactive activities, and numerous practice questions. It allows students to map their performance and check their preparation level.
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• Students can sometimes miss their classes, so they may have some questions about the curriculum. Through Extramarks, students can easily clear their doubts by interacting with their teachers live. Students can also refer to the Equation Of A Line Formula to clear their doubts.
• Extramarks provides students with performance reports so that they can easily track their preparation and progress.

Point Slope Form of Equation of Line

Point-slope equations require a point on the line and its slope. A line has a slope of m. A point represents the x- and y-coordinates of the point, and the slope m is the inclination of the line with the positive x-axis. It is possible for m to have a positive slope, a negative slope, or a zero slope.

Two Point Form of Equation of Line

During this lesson, one will discuss the equation for a line that passes through two points, say (x1, y1) and (x2, y2).  A two-point equation of a line is known as the two-point form. Apply the point-slope method to derive the equation. Assume the slope of the line is m. As a result, the Equation Of A Line Formula will be:

In other words: y – y1 = m(x – x1) …I

However, one does not know the value of m. The line passes through another point (x2, y2).

Therefore, the coordinates (x2, y2) must satisfy the equation above.

y2 – y1 = m(x2 – x1)

⇒ The formula for m is (y2 – y1)/(x2 – x1)

That is it -found the slope. On substituting this in equation (I), one gets the required equation as

y – y1 = {(y2 – y1)/(x2 – x1)}(x – x1)

Also, if two points are (x1, y1) and (x2, y2), then the slope of the line connecting them equals:

(y2 – y1)/(x2 – x1)

AB slope is equal to AC/BC or (y2 – y1)/(x2 – x1). It will be quite useful to have this expression on hand.

Slope Intercept Form of Equation of Line

Lines have slope-intercept forms y = mx + c. The line’s slope is m and its y-intercept is c. The y-axis is cut by this line at (0, c), and c is its distance from the origin. Mathematics and engineering use the slope-intercept form of the Equation Of A Line Formula extensively.

y = mx + c

Intercept Form of Equation of Line

In intercept form, a line’s equation is defined by its x-intercept ‘a’ and y-intercept ‘b’. Line (a, 0) intersects the x-axis and y-axis at point (0, b), and a, b are the distances between these points. This intercept form of the Equation Of A Line Formula can be obtained by substituting these two points in the two-point form of the Equation Of A Line Formula. The intercept form explains where the line intersects the x-axis and y-axis.

Equation of a Line Using Normal Form

Using the perpendicular of the line, which passes through the origin, one can obtain the normal form of the Equation Of A Line Formula. This line that passes through the origin and is perpendicular to the given line is called the normal. Using the parameters of the normal ‘p’ and its angle with the positive x-axis, one can form the Equation Of A Line Formula. Line equations have the following normal form:

xcosθ + ysinθ = P

Additionally, in addition to the above-defined forms of the Equation Of A Line Formula, one can also use the equation of line calculator to conveniently find the equation of a line quickly and easily. For the Equation Of A Line Formula calculator to work, one must provide the slope m and y-intercept c, so that one can obtain the slope-intercept form and standard form of the equation of a line.

How to Find Equation of Line?

Based on the data one has, one can apply the formulas for any of the forms explained above to find the equation of a line. Following are the steps that can be followed for different cases, depending on the parameters and the form.

• The first step is to note down the provided data, the slope of the line as ‘m’ and the coordinates of the point(s) in the form (xn, yn).
• Apply the required formula based on the given parameters, (i) For finding the Equation Of A Line Formula given its slope or gradient and its intercept on the y-axis – slope intercept form.
  (ii) Using a slope and a coordinate of one point on a line to find the Equation Of A Line Formula.
  A two-point form can be used to find the Equation Of A Line Formula given the coordinates of two points lying on it.
  The x-intercept and y-intercept can be used to write an equation with the x-intercept and y-intercept as parameters.
• To express the Equation Of A Line Formula in standard form, rearrange the terms.

The alternative method for cases (ii), (iii), and (iv) might be to first calculate the slope by applying the slope formula to the given data, and then finally to apply the slope-intercept formula.

Equation of Horizontal and Vertical Line

In general, x = a, where a is the y-coordinate of any point lying on the line, can be used to find the equation of a horizontal or parallel line. Lines parallel to the y-axis can be written as y = b, where b is the x-coordinate of any point on the line.

• Y = 0 is the equation of the x-axis, and x = 0 is of the y-axis.
• Lines parallel to the x-axis have the equation y = b, as they cut the y-axis at point (0, b).
• A line parallel to the y-axis has the equation x = a, and it cuts the x-axis at point (a, 0).
• A line parallel to ax + by + c = 0 has the equation ax + by + k = 0.
• A line perpendicular to ax + by + c = 0 has the equation bx – ay + k = 0.

Examples on Equation of Line

In Example 1, what is the normal form of the Equation Of A Line Formula?

Solution:

A normal with length P is inclined at an angle θ with the positive x-axis.

The projection of normal on x-axis and y-axis is Pcosθ and Psinθ respectively.

P has coordinates (Pcosθ, Psinθ).

The slope of normal is tanθ, and the slope of required line perpendicular to the normal is -1/tanθ
Now the point is (Pcosθ, Psinθ), and the required slope m = -1/Tanθ to form the Equation Of A Line Formula.
(y – Psinθ) = -1/tanθ. (x – Pcosθ)
(y – Psinθ) = -1/sinθ/cosθ. (x – Pcosθ)
(y – Psinθ) = -cosθ/sinθ. (x – Pcosθ)
sinθ(y – Psinθ) = -cosθ. (x – Pcosθ)
ysinθ – Psin2θ = -xcosθ +Pcos2θ
xcosθ + ysinθ = Psin2θ + Pcos2θ
xcosθ + ysinθ = P(sin2θ + cos2θ)
xcosθ + ysinθ = P
Hence, the expression for the normal Equation Of A Line Formula is proved.

Example 2: Find the Equation Of A Line Formula with an x-intercept of 5 units and a y-intercept of 4 units. Represent this equation in standard form as well.
Solution:
The x-intercept is a = 5, and y = 4.
Applying this to the intercept form of the equation of a line x/a + y/b = 1, we have the Equation Of A Line Formula as follows.
x/5 + y/4 = 1
Further, the equation is converted into standard form.
x/5 + y/4 = 1
(4x + 5y)/20 = 1
4x + 5y = 20
4x + 5y – 20 = 0
Therefore, 4x + 5y = 20 is the standard form of the Equation Of A Line Formula.

Practice Questions

1. Find the slope and y-intercept of the line with equation 3x – 4y + 7 = 0.

Solution:
The given Equation Of A Line Formula is 3x – 4y + 7 = 0
This equation needs to be converted in slope intercept form of the Equation Of A Line Formula.
3x – 4y + 7 = 0
3x + 7 = 4y
4y = 3x + 7
y = 3x/4 + 7/4
Comparing this equation with the slope-intercept form of the equation of line y = mx + c we have the slope m = 3/4, and the y-intercept c = 7/4.
7/4. Students can have access to a number of practice questions on the Extramarks website.