The slope-intercept concept is related to the representation of the equation of a straight line. It is a form of writing the equation of a straight line where the slope (the inclination of the line with the x-axis) and y-intercept (where the line crosses the vertical y-axis) are known or given. To represent a straight line in the slope-intercept form, we have to know the value of the slope of the line and the intercept value where the line cuts the y-axis. This topic can be learned in an interesting way on Cuemath.
In the slope-intercept form, an equation of a straight will be as follows:
y = mx + b
Where m denotes the slope of the line, b denotes the y-intercept of the line, and (x, y) denotes any point on the line. Here x and y are known to be variables as they can take any values that represent the coordinate of any point on that line.
For example, the slope-intercept form of linear equations is given as follows:
- y = 3x + 4 (Here 2 is the slope and 4 is the y-intercept of the line)
- y = -x + 2.5 (Here -1 is the slope and 2.5 is the y-intercept)
Any other form of the linear equation can be converted to the slope-intercept form to verify its nature. This means if any equation can be expressed in the slope-intercept form, then the given equation can be considered as a linear equation representing a straight line.
For example, the equation 2x + 5y -3 =0 can be rearranged and written as y = -2/5x +3/5 which is in the slope-intercept form and indicates a straight line having a slope of (-2/5) and y-intercept as 3/5
The slope-intercept form is one of the most common ways to represent the equation of a straight line.
All the points that satisfy the equation will lie on this line. A line that passes through the origin (0,0) has a y-intercept as 0 so the slope-intercept form will be in the form y = mx + 0. Again, a line that is horizontal to the x-axis has the slope 0 so the slope-intercept form will be y= 0 + b.
Point Slope Form
The point-slope form is the method of writing an equation that represents a straight line. When the slope of the line (where the line intersects the vertical y-axis) is given, and coordinates of any point on the line (x1, y1) are known, the point-slope form is used to write the equation of the line.
Point slope form is used to represent a straight line by using the value of its slope, and the x-coordinate and y-coordinate of any point on that line. When the slope of a line is given as ‘m’ and the line passes through a point (x1, y1), then the point-slope form of the line is given as follows:
y – y1 = p (x – x1)
Where,
- p = the slope of the line.
- x and y are variables and indicate any random value of coordinates for any point on the line.
- x1 and y1 are the coordinates of a fixed point on the line.
In the case of the slope-intercept form of writing an equation, the value of slope and y-intercept is known. To present an equation in point-slope form, the value of slope and coordinates of any point on the line are required.
For a horizontal line parallel to the x-axis, the slope m is zero, so the point-slope form of the equation takes the form of y-y1=0 or y = y1 where any value of y1 represents the y-intercept.
The point-slope form is also called the point-gradient form. It is a very useful form to represent a straight line when a single point on the line and the slope of the line is known.
