Question

How do you find the slope and intercept of $ 2x + 3y = 6 $ ?

Answer 1

Hint: Change of the form of equation will give us the slope of the line $ 2x + 3y = 6 $ . We have to change it to the form $ y = mx + c $ to find the slope $ m $ . Then, as we know that there are two kinds of intercepts which are $ x $ -intercept and $ y $ -intercept. So, $ x $ -intercept is the point where the line intersects the $ x $ -axis and $ y $ -intercept is the point where the line intersects the $ y $ -axis. So, to calculate the intercepts, we will put $ x $ and $ y $ as zero one by one.

Complete step-by-step answer:
(i)
We are given the line equation:
 $ 2x + 3y = 6 $
In order to find the slope of the line we will have to convert this equation into slope-intercept form i.e.,
 $ y = mx + c $
Therefore, we will subtract $ 2x $ from both the sides of the equation:
 $ 2x + 3y - 2x = 6 - 2x $
On simplifying, it will become:
 $ 3y = 6 - 2x $
Now, we will divide both the sides of the equation by $ 3 $ :
 $ \dfrac{{3y}}{3} = \dfrac{{6 - 2x}}{3} $
On simplifying, we will get:
 $
  y = \dfrac{6}{3} - \dfrac{{2x}}{3} \\
  y = 2 - \dfrac{2}{3}x \;
  $
Writing the equation in slope intercept form, it will look like:
 $ y = - \dfrac{2}{3}x + 2 $
Now, since we have our equation in the slope-intercept form, we will compare the above equation with $ y = mx + c $ to find the value of $ m $ .
As we can see that the coefficient of $ x $ is $ m $ , in our equation the coefficient of $ x $ is $ - \dfrac{2}{3} $ .
i.e.,
 $ m = - \dfrac{2}{3} $
Therefore, the slope of the equation $ 2x + 3y = 6 $ is $ - \dfrac{2}{3} $
So, the correct answer is “ $ - \dfrac{2}{3} $ .”.

(ii)
Now, as we know that $ x $ -intercept is the point where the line crosses the $ x $ -axis and we also know that on $ x $ -axis, $ y = 0 $ . Therefore, to find the $ x $ -intercept, we will put $ y $ as $ 0 $ in the equation of line given to us. Therefore,
 $
  2x + 3\left( 0 \right) = 6 \\
  2x = 6 \\
  x = \dfrac{6}{2} \\
  x = 3 \;
  $
So, the correct answer is “ x = 3 .”.

Therefore, the $ x $ -intercept of the equation $ 2x + 3y = 6 $ is $ 3 $ .
(iii)
Similar to $ x $ -intercept, $ y $ -intercept is the point where the line crosses the $ y $ -axis and we also know that on $ y $ -axis, $ x $ =0. Therefore, to find $ y $ -intercept, we will put $ x $ as $ 0 $ in the equation of the line given to us. Therefore,
 $
  2\left( 0 \right) + 3y = 6 \\
  3y = 6 \\
  y = \dfrac{6}{3} \\
  y = 2 \;
  $
Therefore, the $ y $ -intercept of the equation $ 2x + 3y = 6 $ is $ 2 $ .
So, the correct answer is “ $ 2 $ .”.

Note: A line parallel to $ x $ -axis, does not intersect the $ x $ -axis at any finite distance and hence, we cannot get any finite $ x $ -intercept of such a line. Slope of such a line is $ 0 $ . Similarly, lines parallel to the $ y $ -axis, do not intersect $ y $ -axis at any finite distance and hence, we cannot get any finite $ y $ -intercept of such a line. Slope of such a line is $ \infty $ .