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How do you write the standard form of a line given x intercept $ = 3 $ , y intercept $ = 2 $ ?

Last updated date: 26th Feb 2024
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IVSAT 2024
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Hint: We are given a line whose x intercept and y-intercept is given. Here the x-intercept means $ y = 0 $ and y-intercept means $ x = 0 $ .
X intercept means $ \left( {x,0} \right) $ and y-intercept means $ \left( {0,y} \right) $ . As we got two coordinates then we can find the slope of the line using the formula.
\[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Here \[m\]is the slope, \[{y_2}\& {\text{ }}{y_1}\]are the\[\;y\] coordinates.\[{x_1}and{\text{ }}{x_2}\]are the \[x\]coordinates.
Then using the slope intercept form of the equation of a line i.e.\[y = mx + b\].
Here \[m\]is the slope, \[b\]is the y-intercept. Then using the values we can substitute in the equation and form the standard equation of the line.

Complete step by step answer:
We are given\[x\]intercept $ = 3 $ and \[y\]intercept $ = 2 $ of a line. We have to find its standard form of the equation.
Here x-intercept means $ \left( {x,0} \right) $ i.e. $ \left( {3,0} \right) $ and y-intercept means $ \left( {0,y} \right) $ i.e. $ \left( {0,2} \right) $ . Hence we get two coordinates now using the formula of slope first we will find the slope of the line.

Using the formula of slope \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Here $ {y_1} = 0,{y_2} = 2,{x_1} = 3,{x_2} = 0 $ on substituting these values in the formula we will get:
 $ \Rightarrow m = \dfrac{{\Delta y}}{{\Delta x}} = \dfrac{{2 - 0}}{{0 - 3}} $
On further solving we will get:
 $ \Rightarrow m = \dfrac{{ - 2}}{3} $
Then we will use the slope intercept form of the equation of a line is:
\[y = mx + c\]
Here $ m = \dfrac{{ - 2}}{3};c = 2 $
Therefore on substituting the value in the equation. Therefore the equation of this line can be written as:
 $ \Rightarrow y = \dfrac{{ - 2}}{3}x + 2 $
Multiplying both the sides by \[3\]to clear the fraction the equation can be re-written as:
 $ \Rightarrow 3y = 2x + 6 $
Hence the standard equation is $ 3y = 2x + 6 $

Note: In such type questions mainly get confused by reading the word intercepts. Here given x- intercept and y- intercept should be converted into the coordinate form. By using that coordinate form we can easily solve the whole question. If only one intercept i.e. x-intercept is given then by substituting \[x\] equals to zero in the equation we can find the value of\[y\]intercept or vice versa and then from the equation.