Question

Write the equation of a line whose x-intercept is 3 and the y-intercept is 2.

Answer 1

Hint: We know that the equation of a slope is given by $ m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} $ .
Here $ m $ represents slope and $ \left( {{x_1},{y_1}} \right) $ , $ \left( {{x_2},{y_2}} \right) $ represent the two points. So first we need to find the slope for which we need to find the points using the given information and substitute it in the equation given above.
Also the point slope formula states that $ \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right) $
In this equation, we have to substitute the slope that we found and thus find the equation of the line.

Complete step by step answer:
Given
x intercept: 3 $ \Rightarrow $ x intercept of 3 is the point $ \left( {3,0} \right) $
y intercept: 2 $ \Rightarrow $ y intercept of 2 is the point $ \left( {0,2} \right) $
So here $ \left( {{x_1},{y_1}} \right) $ is $ \left( {3,0} \right) $ and $ \left( {{x_2},{y_2}} \right) $ is $ \left( {0,2} \right) $
Now using the first equation to find the slope of the line such that $ m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} $ :
\[
   \Rightarrow m = \dfrac{{2 - 0}}{{0 - 3}} \\
   \Rightarrow m = - \dfrac{2}{3}...............\left( i \right) \\
 \]
Now on getting $ m $ , substituting the value of $ m $ in point slope formula which is $ \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right) $
Also here $ \left( {{x_1},{y_1}} \right) $ is $ \left( {3,0} \right) $ .
 $ \Rightarrow \left( {y - 0} \right) = - \dfrac{2}{3}\left( {x - 3} \right)...............\left( {ii} \right) $
Solving (ii) to get the equation of the line.
\[
   \Rightarrow y = - \dfrac{2}{3}\left( {x - 3} \right) \\
   \Rightarrow y = - \dfrac{2}{3}x + \left( {\dfrac{2}{3} \times 3} \right) \\
   \Rightarrow y = - \dfrac{2}{3}x + 2..................\left( {iii} \right) \\
 \]
Therefore the required equation is represented in (iii):
\[y = - \dfrac{2}{3}x + 2\]
Alternative method:
By using the slope intercept equation of a line we can directly find the equation of the line.
The slope intercept equation of a line is: $ y = mx + b $
Here $ m $ is the slope and $ b $ is the y-intercept.
So substituting the given values and values in (i) we get:
 $
  \;\,\;\;\;y = mx + b \\
   \Rightarrow y = - \dfrac{2}{3}x + 2 \\
  $
So by using this method one can directly find the equation using simple substitution.

Note:
While doing this type of questions always consider the following points:
x intercept of \[a\] is the point $ \left( {a,0} \right) $
y intercept of \[b\] is the point $ \left( {0,b} \right) $
Point slope formula mainly connects the s