Equations of a Straight Line

Equations of a straight line are linear equations. Before getting into straight-line equations, let’s define a straight line. A straight line never ends and is one-dimensional with no width. It is made up of infinite points connected on both edges of a point. A straight line is free of any curves. It can be slanted, vertical, or horizontal. We get a 180-degree angle if we construct an angle among any two points on a straight line. A straight line has no bends and is infinite in length. It can be drawn between two points, but both endpoints must reach infinity. 

Equations of a Straight Line Meaning

The meaning of straight-line equations is the relationship between any points between x and y coordinate lines. A straight line possesses distinct forms depending on the cartesian plane’s constants, variables, and angles. 

The slope formula or rise over run is as follows:

Slope of a Straight Line = (y Co-ordinates Difference / x Co-ordinates Difference)

The general straight-line equation is expressed as:

General Straight Line Equation: ax + by + c = 0

Where x and y denote variables, and a, b, and c represent constants.

The Gradient of a Straight Line

The gradient of a straight line or the slope of a straight line is the tangent to an angle. A straight line shall have a slope 

m = tan θ. 

In the above slope, θ is the angle formed by the positive x-axis line, and the y-intercept is expressed as:

y = (mx) + c 

Where y denotes the intercept, c represents the y value when x is 0, and m represents gradient or slope.

How to Find the Slope?

Slope Point Form:

In slope point form, a straight line possessing a slope tanθ that is passing through (x1,y1) points is

(y – y1) = (x – x1) * (m)

2 Points Form:

When 2 points are present: (x1,y1)  and (x2,y2) 

Slope or Gradient or m = (Change in y value) / (Change in x value) 

Therefore, slope or Gradient or m = (y2 – y1) / (x2 – x1) 

Equations of a Straight Line Questions and Solutions

Example 1:

Imagine a straight line y = 3x + 9, we shall find the slope and y-intercept.

The slope or gradient (m) of a straight line shall be calculated using y = (mx) + c.

Comparing y = 3x + 9 to the above equation, we shall answer slope or m = 3 and c or y-intercept = 9.

Example 2:

A gated community is located on a cartesian plane, Lara’s home is located at (4, 3), and Dhoni’s home is (7, -2). Two routes must be drawn from a square (3, 2). We must now identify if the two routes are perpendicular.

First, let us imagine that the two routes are straight lines. 

We shall assume Lara’s home is at a point A (4, 3)

And Dhoni’s home is situated at point B(7, -2)

Square seems to be at point C (3, 2)

Apply the formula for gradient (m) between two points, which is

m = (y2 – y1) / (x2 – x1)

The line’s slope between C and A is m1

m1 = (3 – 2) / (4 – 3) 

m1 = 1 / 1 

m1 is 1

The line’s slope between B and C is m2

m2 = (-2 – 2) / (7-3)

m2 = – 4 / 4

m2 = -1

If two routes seem perpendicular, the slope’s product should be -1.

It means m1 * m2 = 1 * -1

m1 * m2 = -1

As the slope’s product is -1, the two routes seem to be perpendicular.

Example 3:

Let us find the straight line equation that shall go through the points (3, 4) and (1, 2).

The equations of a straight line crossing 2 points is:

y = (mx) + c

m = (y2 -y1) / (x2 – x1) = (2 – 4) / (1 – 3) = 1

Using (2, 4), we shall get

4 = (1 * 2) + ( c ) 

c + 2 = 4

c = 2

Hence, y = x + 2 is the answer.

Conclusion

The meaning of straight-line equations is the relationship between any points between x and y coordinate lines. It can take several forms. A straight line possesses distinct forms depending on the cartesian plane’s constants, variables, and angles. The gradient of a straight line or the slope of a straight line is the tangent to an angle. A straight line shall have a slope m = tan θ in which θ is the angle formed by the positive x-axis line and y-intercept.