Equation of Straight Line

The general equation of a straight line is y = mx + c, where m is the slope of the line and c is the y-intercept. It is the most common form of the equation of a straight line that is used in geometry. The equation of a straight line can be written in different forms such as point-slope form, slope-intercept form, general form, standard form, etc. A straight line is a two-dimensional geometrical entity that extends on both its ends till infinity.

In this article, we will explore the concept of the equation of a straight line. We will try to understand the general equation of a line, straight-line formula, the way of finding the equation of a straight line, and discover other interesting aspects of it. Try your hands at solving a few interesting examples and questions for a better understanding of the concept.

The equation of a straight line is a mathematical equation that gives the relation between the coordinate points lying on that straight line. It can be written in different forms and tells the slope, x-intercept, and y-intercept of the line. The most commonly used forms of the equation of straight line are y = mx + c and ax + by = c. Some other forms are point-slope form, slope-intercept form, general form, standard form, etc. Let us go through the formula for the equation of a straight line:

Equation of Straight Line Formula

A straight line is a figure formed when two points A (x₁, y₁) and B (x₂, y₂) are connected with minimum distance between them, and both the ends extended to infinity. The standard form of a linear equation with variables x and y is:

ax + by = c, where a, b, c are constants and x, y are variables.

Suppose a line l makes an angle of θ with a positive direction of the x-axis. The angle θ is called the inclination of the line and tan θ is called the slope of the line. Note that x-axis has a slope 0. In fact, all lines parallel to the x-axis have a slope of 0. Also, the slope of all the vertical lines including the y-axis is not defined. Now that we know the basic form of the equation of a line, let us go through different forms of equations of a straight line:

Standard Form of Equation of Line

The standard form of a straight line is given by ax + by = c, where a, b, c are real numbers. Let us consider an example to transform the equation y = 2x - 1 in the standard form. Subtract 2x from both sides of the equation, we have

y - 2x = 2x - 1 - 2x

⇒ y - 2x = -1

⇒ 2x - y = 1

So, we obtain the standard form of the equation of the line as 2x - y = 1.

Point-Slope Form

The equation of a straight line whose slope is m and which passes through a point (x₁, y₁) is found using the point-slope form. The equation of the point-slope form is:

y - y₁ = m (x - x₁), where (x, y) is an arbitrary point on the line.

Let us see how to find the point-slope form. We will derive this formula using the equation for the slope of a line. Let us consider a line whose slope is m. Let us assume that (x₁, y₁) is a known point on the line. Let (x, y) be any other random point on the line whose coordinates are not known. We know that the equation for the slope of a line is:

Slope = Difference in y-coordinates/Difference in x-coordinates

⇒ m = (y - y₁)/(x - x₁)

Multiplying both sides by (x - x₁),

m (x - x₁) = (y - y₁)

This can be written as,

(y - y₁) = m (x - x₁)

Hence the point-slope form of the equation of a line is proved.

Slope-Intercept Form

Now, suppose a line is given to you with its slope m and its y-intercept. Say, a line intersects the y-axis at the point (0, c). Using the point-slope form we have y - c = m (x - 0) ⇒ y = mx + c, where c is the y-intercept. Similarly, if d is the x-intercept, then the slope-intercept form of the equation of the line is y = m(x - d).

The graph of a linear equation in one variable x forms a vertical line parallel to the y-axis and the graph of the equation of a straight line in one variable y is a horizontal line parallel to the x-axis The graph of a linear equation in two variables x and y forms a straight line.

Important Notes on Equation of Straight Line

• The equation of a straight line is also called a linear equation.

• If the product of slopes of two straight lines is -1, then lines are perpendicular to each other.

• If two straight lines are parallel to each other, then they have the same slope.

• Point Slope Form: (y - y₁) = m (x - x₁)

• Slope-Intercept Form: y = mx + c

• Standard Form = ax + by = c

Related Topics on Equation of Line

• Straight Line
• Lines
• Parallel Lines
• Point Slope Form

What is the Equation of Straight Line in Coordinate Geometry?

The equation of a straight line is a mathematical equation that gives the relation between the coordinate points lying on that straight line. The general equation of a straight line is y = mx + c, where m is the slope of the line and c is the y-intercept.

What is a Straight Line in Math?

A straight line is a two-dimensional figure formed when two points A (x₁, y₁) and B (x₂, y₂) are connected with minimum distance between them, and both the ends extended to infinity.

What is the Formula for the Equation of a Straight Line?

Generally, we represent the equation of a straight line using the formula: ax + by = c, where x and y are variables.

What is the y-intercept Formula?

If a line intersects the y-axis at the point (0, c), then c is the y-intercept.

How Do you Find the Slope and Y-intercept of an Equation of a Line?

The slope m of a line joining two points (x₁, y₁) and (x₂, y₂) is given by m = (y - y₁)/(x - x₁). The y-intercept can be found using the equation by substituting the value of x as 0 into the equation of the straight line and finding the corresponding value of y.

The y-intercept of a straight line ax + by = c is found by substituting the x as 0.

How Do you Graph a Slope-Intercept Equation?

The slope-intercept form is given by y = mx + c. It is plotted on a graph by plotting the y-intercept c and then making a straight line passing through (0, c) with slope m.

What is the y-intercept in y = -2x?

On comparing y = -2x with the slope intercept form of equation of a straight line y = mx + c, , we have c = 0. Hence, the y-intercept in y = -2x is 0.

How Do you Write an Equation for a Vertical and Horizontal Line?

The equation of a horizontal line passing through (a, b) is of the form y = b. The equation of a vertical line passing through (a, b) is of the form x = a.

How To Find the Equation of a Straight Line when Given Two Points?

The equation of a straight line joining two points (x₁, y₁) and (x₂, y₂) is given by y - y₁ = (x - x₁)[(y₂ - y₁)/(x₂ - x₁)]