These forms of straight-line
equations are used frequently to represent the line equation in different ways.
Point slope form and slope-intercept form are the most prominent ways to write
a linear equation of a line.
What is the equation of a line?
The equation of a line is a single representation of numerous points on a line that makes up
the equation of that line. An equation of a line can be written as ax + by + c
= 0, which satisfies any point on the line. A slope and a point on a line are
the two minimum requirements for forming an equation for a line.
Forms of the equation of a line
There are various forms of
the equation of the line such as point-slope form, slope intercept form,
standard form of an equation, and x & y-intercept form. We’ll discuss the point-slope
form and slope-intercept form in detail.
Point slope form
The point-slope form is a
well-known method for representing the line equation with the help of
coordinate points of the line. The slope and coordinate points of the line are
the essential part of the point-slope form to represent a line equation.
The slope of a line is the
measure of the tangent line (steepness) by using the x and y coordinate points
of the line. It will be evaluated with the help of a formula such as the slope
of a line is equal to the change in the values of the y-coordinate divided by
the change in the values of the x-coordinate.
The slope of a line can be
positive, negative, zero, or undefined depending on the number of coordinate
points. The expression of a point-slope form is:
y – y1 = m (x – x1)
x and y are the fixed points
on the line, x1 and y1 are the coordinate points of the
line, and m is the slope of a line.
Let us take a few examples
of point slope form to learn how to represent a line equation in the form of a point-slope.
Example 1: For two points
If the coordinate points of
the line are (x1, y1) = (4, 5) & (x2, y2)
= (22, 23), then find the point-slope form to express the linear equation of a
line.
Solution
Step 1: First of all, take
the given coordinate points of the x-axis and y-axis.
x1 = 4, x2
= 22, y1 = 5, y2 = 23
Step 2: Take the given points of the
line to evaluate the slope of a line by placing them in the general formula.
Slope of the line = m = (y2
– y1) / (x2 – x1)
Slope = m = (23 – 5) / (22 –
4)
Slope = m = (18) / (18)
Slope = m = 18/ 18
Slope = m = 9/ 9
Slope = m = 3/3 = 1
Step 3: Now write the
general form of the point-slope form.
y – y1 = m (x – x1)
Step 4: Now substitute the slope and
a pair of points to the general expression of the point-slope form to evaluate
the line equation.
y – y1 = m (x – x1)
y – (5) = 1 * (x – 4)
Required equation of a line.
The line equation through
point-slope form can also be simplified further to express the line equation
prominently.
y – (5) = 1 * x – 1 * 4
y – 5 = x – 4
y – 5 - x + 4 = 0
y – x – 1 = 0
x – y + 1 = 0
Example 2: For 1 point and slope
If the coordinate point of
the line is (x1, y1) = (10, 20) and the slope of a line
is -5, then evaluate the line equation through point slope form.
Solution
Step 1: First of all, take
the given point and the slope of a line.
Slope of a line = m = -5
x1 = 10
y1 = 20
Step 2: Now write the
general form of the point-slope form.
y – y1 = m (x – x1)
Step 3: Now substitute the slope and
a pair of points to the general expression of the point-slope form to evaluate
the line equation.
y – y1 = m (x – x1)
y – 20 = -5 * (x – 10)
Required equation of a line.
The line equation through the
point-slope form can also be simplified further to express the line equation
prominently.
y – 20 = -5 * x + 5 * 10
y – 20 = -5x + 50
y – 20 + 5x – 50 = 0
y + 5x – 70 = 0
5x + y – 70 = 0
A point slope form calculator
is a helpful resoure to determine the equation of the line in the form of point
slope form.
Slope intercept form
Slope intercept form is the
other form of the line equation to represent the equation of a straight line. This
method of a line equation requires a slope of a line and the y-intercept of the
line.
The slope of a line is the
measure of the steepness of the line and the y-intercept of the line is a point
that cuts the y-axis on any point. The slope of a line can be calculated with
the help of rise over run formulas while the y-intercept can be calculated with
the help of slope and a point of the line.
The equation of a line can
be evaluated with the help of slope and y-intercept or two points of the line.
If the slope of a line is not given, then you have to evaluate it with the help
of the rise over run formula.
The general expression of
the slope-intercept form is:
y = mx + b
where x and y are the fixed
points of the line, m is the slope of the line, and b is the y-intercept of the
line.
Let us take a few examples
of slope intercept form to learn how to represent a line equation in the form
of slope intercept.
Example 1: For two coordinate points
If the coordinate points of
the line are (x1, y1) = (1, 9) & (x2, y2)
= (31, 30), then find the slope-intercept form to express the linear equation
of a line.
Solution
Step 1: First of all,
write the points of the line.
x1 = 1, x2
= 31, y1 = 9, y2 = 30
Step 2: Now use the rise
over run formula to calculate the slope of a line by placing the points of x
and y coordinates.
A general expression of the
slope of the line
Slope of the line = m = [y2
– y1] / [x2 – x1]
Put the given values
Slope = m = [30 – 9] / [31 –
1]
Slope = m = [21] / [30]
Slope = m = 7/10
Slope = m = 0.7
Step 3: Now put the slope
and a pair of points to the general expression of the slope-intercept form to
calculate the y-intercept of the line.
y =
mx + b
9
= 7/10(1) + b
9
= 7/10 + b
9
* 10 = 7 + b
90
– 7 = b
b
= 83
Step 4: Now place the slope of a
line and the y-intercept of a line to the general expression of the slope-intercept
form to express the linear equation of the line.
y =
mx + b
y = 0.7x + 83
Example 2: For 1 point & slope
If the coordinate point of
the line is (x1, y1) = (4, 3) and the slope of a line is
-12, then evaluate the line equation through the slope-intercept form.
Solution
Step 1: First of all,
take the given point and the slope of a line.
x1 = 4
y1 = 3
m = -12
Step 2: Now put the slope
and a pair of points to the general expression of the slope-intercept form to
calculate the y-intercept of the line.
y =
mx + b
3
= -12(4) + b
3
= -48 + b
3 +
48 = b
51
= b
Step 3: Now place the slope of a
line and the y-intercept of a line to the general expression of the slope-intercept
form to express the linear equation of the line.
y =
mx + b
y = -12x + (51)
y = -12x + 51
Sum Up
The slope-intercept form and the point-slope form are two well-known methods of the equation of the line. Both methods are totally dependent on the slope of a line and the coordinate points of the line.
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