Question

How do you find the equation of the line passing through the points \[\left( {1,4} \right)\] and \[\left( { - 2,6} \right)\]?

Answer 1

Hint: Here we will find the slope of the line of the two points using the gradient formula. Then by using any one of the points we will find the value of the y-intercept by substituting the points in the slope-intercept form. Finally, we will substitute the value we found in the slope-intercept form to get the desired answer.

Formula Used:
We will use the following formulas:
Slope-intercept form:\[y = mx + b\], where \[m = \] Slope, \[b = \] y-intercept and \[x = \] Independent variable of the function \[y = f\left( x \right)\]
Gradient Formula: \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\], where \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] are the two points given.

Complete step by step solution:
Points given to us are \[\left( {1,4} \right)\] and\[\left( { - 2,6} \right)\]
So, \[\left( {{x_1},{y_1}} \right) = \left( {1,4} \right)\] and \[\left( {{x_2},{y_2}} \right) = \left( { - 2,6} \right)\]
Substituting the above value in the formula \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\], we get,
\[m = \dfrac{{6 - 4}}{{ - 2 - 1}}\]
Adding and subtracting the terms, we get
\[ \Rightarrow m = - \dfrac{2}{3}\]
Now substituting the value of \[m\] in the slope intercept formula \[y = mx + b\], we get
\[y = - \dfrac{2}{3}x + b\]…..\[\left( 1 \right)\]
Now we have to find the value of \[b\]. So, we will substitute any one point in the equation \[\left( 1 \right)\].
Let us substitute \[\left( {1,4} \right)\] in equation \[\left( 1 \right)\], we get
\[4 = - \dfrac{2}{3} \times 1 + b\]
Simplifying the expression, we get
\[ \Rightarrow 4 + \dfrac{2}{3} = b\]
Taking LCM in the LHS and simplifying, we get
\[ \Rightarrow b = \dfrac{{4 \times 3 + 2}}{3} = \dfrac{{14}}{3}\]
So we get \[b = \dfrac{{14}}{3}\].
Substituting the value of \[b\] in equation \[\left( 1 \right)\], we get
\[y = - \dfrac{2}{3}x + \dfrac{{14}}{3}\]

Therefore, \[y = - \dfrac{2}{3}x + \dfrac{{14}}{3}\] is the equation of the line joining points \[\left( {1,4} \right)\] and \[\left( { - 2,6} \right)\].

Note:
A Line is a one-dimensional figure that extends endlessly in both directions. It is also described as the shortest distance between any two points. There are many ways to find the equation of a line such as Point-slope Form, Intercept Form, Determinant Form, and many others. The Form we are using depends on the data we have. We know that in a plane there are an infinite number of points and a number of lines can pass through it.