Question

How to find b in linear equation form $y = mx + b$ if the $2$ coordinates are $(5,6)$ and $(1,0)$?

Answer 1

Hint:
We know the general equation of a straight line is $y = mx + b$ where $m$ is the gradient and $y = b$ is the value where the line cuts the $y - $axis. We know about the cartesian coordinates of points which is $(x,y)$ where $x$ is the abscissa and $y$ is the ordinate.

Complete step by step Solution:
Given that –
Linear equation form $y = mx + b$ and two given coordinates are $(5,6)$ and $(1,0)$
Now we know that if a line passing through two points $({x_1},{y_1})$ and $({x_2},{y_2})$ then equation of line is
$(y - {y_1}) = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_1})$
Now compare given points $(5,6)$ and $(1,0)$ with $({x_1},{y_1})$ and $({x_2},{y_2})$
We get ${x_1} = 5,{y_1} = 6$ and ${x_2} = 1,{y_2} = 0$
Now put all value in the equation line which passing through two points $({x_1},{y_1})$ and $({x_2},{y_2})$ which is $(y - {y_1}) = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_1})$
$ \Rightarrow (y - 6) = \dfrac{{(0 - 6)}}{{(1 - 5)}}(x - 5)$
After calculating all values we get
$ \Rightarrow (y - 6) = \dfrac{{( - 6)}}{{( - 4)}}(x - 5)$
$ \Rightarrow (y - 6) = \dfrac{{( - 1) \times (6)}}{{( - 1) \times (4)}}(x - 5)$
We know that we can multiply and divide any number with $1$ because after multiplication and divide of $1$ we will get same number therefore in above equation we have we have multiply by $1$ with numerator and denomination in left hand side and split negative sign which will cross from each other.
Now after calculating all left hand side numbers we will get $(y - 6) = \dfrac{{(3)}}{{(2)}}(x - 5)$
Now we will multiply both side by $2$then we will get
$ \Rightarrow (2) \times (y - 6) = (2) \times \dfrac{{(3)}}{{(2)}}(x - 5)$
After calculating all numbers, we will get $(2) \times (y - 6) = (3) \times (x - 5)$
Now multiplying both side and we will get
$ \Rightarrow (2y - 12) = (3x - 15)$
Now we will transform or convert it in the form of $y = mx + b$ so we will get our required value of b for this we keep $y$ in the left hand side and all values will transfer to the right hand side
$ \Rightarrow 2y = 3x - 15 + 12$
After calculating all numbers, we will get
$ \Rightarrow 2y = 3x - 3$
For transform or converting it in the form of $y = mx + b$ we have to make coefficient of $y$one for it we will divide both side by $2$then we will get
 $ \Rightarrow y = \dfrac{3}{2}x - \dfrac{3}{2}$
Now compare above equation with the linear equation form $y = mx + b$ then we will get value of b which is our required value
Therefor after comparing both equation we will get $m = \dfrac{3}{2}$ and $b = - \dfrac{3}{2}$

So our required value is $b = - \dfrac{3}{2}$ which is our answer.

Note:
We can solve the above question by directly putting all values in the linear line equation which passes through the two points $({x_1},{y_1})$ and $({x_2},{y_2})$. Another method is the graph method which we can use to find the length of $y = b$ where the line cuts the $y - $axis.