Solution:
As stated in the problem the line is horizontal. This means that its slope is zero.
Mathematically it means that it is a line parallel to the x-axis and passing through y = -3 as shown below as horizontal line 1.
The horizontal line 1 passing through the point (2, -3) is represented by the equation y = -3.
There is another horizontal line 2 which is passing through the point (1,2) which also has slope zero as it is parallel to the x-axis and represented by the equation y = 2
What is the equation of the horizontal line that passes through the point (2, -3)?
Summary:
The equation of the horizontal line that passes through the point (2, -3) is y = -3
Explanation:
Begin by finding the slope of the line using the formula
#m = (y_2-y_1)/(x_2-x_1)#
For the points
#(2,-3) and (1,-3)#
#x_1 = 2#
#x_2 = -3#
#x_2 = 1#
#y_2=-3#
#m = (-3-(-3))/(1-2)#
#m=0/-1#
#m=0#
This equation is actually a horizontal line running through the y axis at #y=-3#
Solution : Let the equation of straight line be `(x)/(a)+(y)/(b)=1`. <br> It passes through the point `(2,3)` <br> `(2)/(a)+(3)/(b)=1`…..`(1)` <br> Given that `a+b=10` <br> `implies b=10-a` <br> From eq. `(1)` <br> `(2)/(a)+(3)/(10-a)=1` <br> `implies (20-2a+3a)/(a(10-a))=1` <br> `implies20+a=10a-a^(2)` <br> `implies a^(2)-9a+20=0` <br> `implies (a-4)(a-5)=0` <br> `implies a=4` or `a=5` <br> If `a=4` then `b=10-4=6` <br> If `a=5` then `b=10-5=5` <br> Therefore, equation of line is <br> `(x)/(4)+(y)/(6)=1` or `(x)/(5)+(y)/(5)=1` <br> `implies 3x+2y=12` or `x+y=5`
First, let's see it in action. Here are two points (you can drag them) and the equation of the line through them. Explanations follow.
../geometry/images/geom-line-equn.js
The Points
We use Cartesian Coordinates to mark a point on a graph by how far along and how far up it is:
Example: The point (12,5) is 12 units along, and 5 units up
Steps
There are 3 steps to find the Equation of the Straight Line :
- 1. Find the slope of the line
- 2. Put the slope and one point into the "Point-Slope Formula"
- 3. Simplify
Step 1: Find the Slope (or Gradient) from 2 Points
What is the slope (or gradient) of this line?
We know two points:
- point "A" is (6,4) (at x is 6, y is 4)
- point "B" is (2,3) (at x is 2, y is 3)
The slope is the change in height divided by the change in horizontal distance.
Looking at this diagram ...
Slope m = change in ychange in x = yA − yBxA − xB
In other words, we:
- subtract the Y values,
- subtract the X values
- then divide
Like this:
m = change in y change in x = 4−3 6−2 = 1 4 = 0.25
It doesn't matter which point comes first, it still works out the same. Try swapping the points:
m = change in y change in x = 3−4 2−6 = −1 −4 = 0.25
Same answer.
Step 2: The "Point-Slope Formula"
Now put that slope and one point into the "Point-Slope Formula"
Start with the "point-slope" formula (x1 and y1 are the coordinates of a point on the line):
y − y1 = m(x − x1)
We can choose any point on the line for x1 and y1, so let's just use point (2,3):
y − 3 = m(x − 2)
We already calculated the slope "m":
m = change in ychange in x = 4−36−2 = 14
And we have:
y − 3 = 14(x − 2)
That is an answer, but we can simplify it further.
Step 3: Simplify
Start with:y − 3 = 14(x − 2)
Multiply 14 by (x−2):y − 3 = x4 − 24
Add 3 to both sides:y = x4 − 24 + 3
Simplify:y = x4 + 52
And we get:
y = x4 + 52
Which is now in the Slope-Intercept (y = mx + b) form.
Check It!
Let us confirm by testing with the second point (6,4):
y = x/4 + 5/2 = 6/4 + 2.5 = 1.5 + 2.5 = 4
Yes, when x=6 then y=4, so it works!
Another Example
Example: What is the equation of this line?
Start with the "point-slope" formula:
y − y1 = m(x − x1)
Put in these values:
- x1 = 1
- y1 = 6
- m = (2−6)/(3−1) = −4/2 = −2
And we get:
y − 6 = −2(x − 1)
Simplify to Slope-Intercept (y = mx + b) form:
y − 6 = −2x + 2
y = −2x + 8
DONE!
The Big Exception
The previous method works nicely except for one particular case: a vertical line:
A vertical line's gradient is undefined (because we cannot divide by 0):
m = yA − yBxA − xB = 4 − 12 − 2 = 30 = undefined
But there is still a way of writing the equation: use x= instead of y=, like this:
x = 2
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