Geometry is described as the study of the measurement, qualities, and connections of points, lines, angles, surfaces, and solids. Simply said, geometry is a branch of mathematics that deals with points, lines, forms, and surfaces. When you hear the word “geometry,” you probably think about forms, area, volume, lines, rays, and vertices and that is exactly what geometry is!
What is a line?
A line is a two-dimensional figure with length but no breadth. A line is made up of a series of points that stretch in opposite directions indefinitely. Two points in a two-dimensional plane determine it. A straight pair of points extending in opposing directions can be characterized as a line. It has no endpoints in any direction (it is limitless). It is one-dimensional and has no thickness.
There are several sorts of lines in geometry, such as horizontal and vertical lines, parallel and perpendicular lines. These lines are crucial in the building of many sorts of polygons. A square, for example, is formed by four lines of equal lengths, but a triangle is formed by linking three lines end to end.
Basic Terminology
- X-axis: In a Cartesian coordinate system, the x-axis refers to the horizontal plane. Perpendicular projections from the x- and y-axes can be used to find a place on a two-dimensional graph. In a two-dimensional system, the first and second coordinates are known as the abscissa and ordinate of P, respectively.
- Y-axis: On the Cartesian coordinate plane, the y-axis is the vertical axis. The y-axis extends from negative infinity to positive infinity. The y-axis also serves as the starting point, or 0 points, for determining how far a point extends horizontally on a graph.
- Slope: The slope of a line is a measurement of its steepness and direction. It is defined as the change in y coordinate with regard to that line’s change in x coordinate.
- Y-intercept: A graph’s y-intercept is the point at which the graph meets the y-axis.
Equation of a line formula
The slope-intercept form of a straight line is used to obtain a line’s equation. The slope of the line and the intercept cut by the line with the y-axis are required for the slope-intercept formula.
The slope-intercept formula is given by,
y = mx + c
where,
m = slope of the line
c = y-intercept of the line
Derivation
Consider two points on the above line (x1, y1) = (0, c) and (x2, y2) = (x, y).
The slope of the line is given by,
mx = y – c
y = mx + c
This is the general equation for a straight line, which includes its slope and y-intercept. As a result, the equation of line formula is derived.
Example: Find the equation of a line with slope 2 and y-intercept 4.
We have, m = 2 and c = 4
The equation of a line is given by slope-intercept form, that is, y = mx + c.
So, the required equation is,
y = 2x + 4
Sample Problems
Question 1. Find the equation of a line with slope –5 and y-intercept 1.
Solution:
We have, m = –5 and c = 1
The equation of a line is given by slope-intercept form, that is, y = mx + c.
So, the required equation is,
y = –5x + 1
Question 2. Find the equation of a line with slope 7 and y-intercept 9.
Solution:
We have, m = 7 and c = 9
The equation of a line is given by slope-intercept form, that is, y = mx + c.
So, the required equation is,
y = 7x + 9
Question 3. Find the equation of a line with slope 1/2 and y-intercept 4.
Solution:
We have, m = 1/2 and c = 4
The equation of a line is given by slope-intercept form, that is, y = mx + c.
So, the required equation is,
y = (1/2)x + 4
y = x/2 + 4
2y = x + 8
Question 4. Find the equation of a line with slope –10 and y-intercept 1/4.
Solution:
We have, m = –10 and c = 1/4
The equation of a line is given by slope-intercept form, that is, y = mx + c.
So, the required equation is,
y = –10x + 1/4
4y = –40x + 1
Question 5. Find the equation of a line with slope 2 and y-intercept –3.
Solution:
We have, m = 2 and c = –3
The equation of a line is given by slope-intercept form, that is, y = mx + c.
So, the required equation is,
y = 2x + (–3)
y = 2x – 3
Question 6. Find the equation of a line with slope 4 and y-intercept –1.
Solution:
We have, m = 4 and c = –1
The equation of a line is given by slope-intercept form, that is, y = mx + c.
So, the required equation is,
y = 4x – 1