In this article, you will learn about the concept of the point-slope form of a line. From the concept of finding the slope of a line when two points are given, the point-slope form is derived. When we can find the equation of the straight line which is inclined at an x-axis angle and passes through a given point, we will use the point-slope form of a line.
On this page, we will discuss the 10th Grade Math concept of the Point-slope form of a line. We will try to make you understand the point-slope form, formulas, how to derive the point-slope form formula, solved examples, and so on. This point-slope formula is only used when we know the slope of the line and a point on the line.
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Point-Slope Form of a Line Definition
The point-slope form is defined as when we find the equation of a line with one point and slope which is inclined at an angle to the positive direction of the x-axis in the anti-clockwise direction and passes through a given point by an equation of a line in point-slope form. This is represented as using slope and point on the line of the straight line. The point-slope form of a line figure is as shown below,
Point Slope Form Formula
In an equation of a line, the slope of a line is ‘m’ and which passes through a point (x1, y1). For finding the equation of a straight line we can express it in different forms such as slope-intercept form, intercept form, normal form, point-slope form, etc. The equation of a straight line using the point-slope form formula is,
(y-y1) = m(x-x1)
- Where (x, y) is a random point on the line.
- m is the slope of the line.
- (x1,y1) is a fixed point on the line.
Point-Slope Form Formula Derivation
Now, we will derive the formula of the equation of a straight line using the point-slope form. Consider that the slope of a line is ‘m’ and the points on the line are (x1,y1), (x,y) be random points on the line whose coordinates are unknown. The point-slope form of a line formula is,
Point-slope form of a line is (y-y1) = m(x-x1).
We know the formula for the slope of a line.
So, the Slope of a line = (Difference in y-coordinates)/(Difference in x-coordinates)
i.e., m = (y-y1)/(x-x1).
Now Multiply both sides by (x-x1). Then we get
m(x-x1) = ((y-y1)/(x-x1)) x (x-x1).
After multiplying, this can be
m(x-x1)= (y-y1)
Finally, rewrite the equation such that the y-variables are on the left side to get to the desired form.
(y-y1) = m(x-x1)
Hence, proved the point-slope form formula.
How do you find the Point-Slope Form of a Line?
The following are the steps to find the equation of the straight line using the point-slope form of a line. The steps are:
Step 1: First, note down the slope of a line ‘m’, and the points that are lies on the line are coordinate points (x1,y1).
Step 2: Now, Substitute the given values in the point-slope formula. The formula is (y-y1) = m(x-x1)
Step 3: Finally, Simplify the equation to get the equation of a line in the standard form.
Read More:
- Coordinate Geometry Graph
- Graph of Standard Linear Relations Between x, y
- Slope of the Graph of y = mx + c
Point-Slope Form Example Problems with Solutions
Problem 1: Find the equation of a line that passes through (3, -4) and with a slope is -5.
Solution:
Given that,
The points (x1,y1) are (3,-4).
Next, the slope of a line is m =-5.
We know the formula of the equation of a line using a point-slope form is,
(y-y1) = m(x-x1)
After substituting the given values in the above formula. It will be,
y-(−4) = (−5)(x − 3)
i.e., y+4 = -5x +15
5x +y+4 -15 =0
5x+y-11 =0
Hence, the equation of the line is 5x+y – 11 =0.
Problem 2: Find the equation of a line that passes through the points (2, –2) and (3, 5) in point-slope form.
Solution:
In the given question,
The points (x1 ,y1) is (2, -2).
The points of (x2,y2) are (3, 5).
The formula for the slope of the line passes through the points is.
m = (y2 – y1)/ (x2 – x1)
After substituting the given values in the above formula. Then we get,
m = (5-(-2))/(3-2)
m= 7/1
So, the slope of the line m is 7.
The equation of a line passing through the point (2, -2) with slope 7.
The formula for equation of a line using point-slope form is,
(y-y1) = m(x-x1)
y -(-2) = 7(x – 2)
y + 2 = 7x – 14
-7x + y + 2 + 14 = 0
-7x + y + 16= 0
Similarly, the equation of a line passing through the point (3, 5) with slope of a line 7.
Substitute these value in a point slope from formula. Then we get,
y – 5 = 7(x – 3)
y – 5 = 7x – 21
-7x + y – 5 + 21= 0
-7x + y + 16 = 0
Therefore, the equation of the line in point slope form is -7x + y + 16 = 0.
Problem 3: A straight line that passes through the point (4, -6) and the positive direction of the x-axis gives an angle of 135 °. Find the equation for a straight line?
Solution:
As given in the question,
The points (x1 ,y1) are (4,-6)
An angle of 135 ° with a positive direction of the x-axis line.
The slope of the line m is,
m= tan 135 ° = tan (90 ° + 45 °) = – cot 45 ° = -1.
The line that is needed to pass through the point (4, -6).
We know the equation of a line using the point-slope formula.
The formula is (y-y1) = m(x-x1)
Substitute the values in the above formula. We get,
y – (-6) = -1 (x -4)
y + 6 = -x + 4
x + y + 6- 4 = 0
x + y + 2 = 0
Thus, the equation of a straight line is x + y + 2 = 0.
Problem 4: Find the slope of the line when the equation of the line is 2x – 3y + 1.
Solution:
Given that, the equation of a line is 2x-3y+1
Now, we will find the value of the slope of a line.
The given equation of a line is in the form of ax + by + c is -a/b
So, the value of a is 2 and the value of the b is -3.
then the value of the m is -(2/-3)
Hence, the slope of a line is 2/3.
Problem 5:Â Find the equation of the line which passes through two points (3,6) and (4,8).
Solution:Â
In the given question,
The points (x1 ,y1) is (3, 6).
The points of (x2, y2) are (4, 8).
The two points are given. So, the two points slope form can be
y-y1 = (y2-y1)/(x2-x1) * (x-x1)
Substitute the values in the above formula as x1 = 3, y1 = 6, x2 = 4, y2 = 8
then the equation is y -6 = (8-6 )/(4-3) * (x-3)
y-6 = 2*(x-3)
y -6 = 2x-6
2x-y =0
Hence, the equation of the line is 2x-y = 0
FAQ’s on Point-Slope Form of a Line
1. How do you find the slope for two points?
The two points (x1, y1) and (x2, y2) be given, then the slope of the line passing through these points is: m = (y2 – y1)/(x2 – x1)
2. When can we use the point-slope of a line formula?
The formula of the point-slope form of an equation is used when we need to find the equation of a straight line given a point on it and the slope of the line.
3. How do you Change Point-Slope Form into the Slope-Intercept Form?
The formula for the point-slope form of a line is (y-y1) = m(x-x1). We will solve this equation for y which gives an equation in the form is y = mx + b. This is called the slope-intercept form.
4. What are the Applications of the Point-Slope Formula?
The applications of the point-slope of a line formula are,
- We can find the equation of a line with the given slope and a point on it.
- It can be used to Graph the line by using the slope and one point on the line.
- One can find the slope of the line right away from the equation of the line.