Question

How do you graph $ 3x - 5y = 15 $ using intercepts?

Answer 1

Hint: 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 and then once we get both the intercepts, we will use the coordinates of the intercepts and draw the line on a graph.

Complete step-by-step answer:
(i)
We are given the line equation:
 $ 3x - 5y = 15 $
As we are asked to draw the graph of $ 3x - 5y = 15 $ using intercepts, we first need to calculate both of the intersects namely, $ x $ -intercept and $ y $ -intercept.
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,
 $
  3x - 5\left( 0 \right) = 15 \\
  3x = 15 \\
  x = \dfrac{{15}}{3} \\
  x = 5 \;
  $
Therefore, the $ x $ -intercept of the equation $ 3x - 5y = 15 $ is $ 5 $ .
(ii)
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,
 $
  3\left( 0 \right) - 5y = 15 \\
   - 5y = 15 \\
  y = - \dfrac{{15}}{5} \\
  y = - 3 \;
  $
Therefore, the $ y $ -intercept of the equation $ 3x - 5y = 15 $ is $ - 3 $ .
(iii)
Now, to draw a graph we need two points which lie on the line. As we have calculated both the intercepts, we can say that the line crosses the $ x $ -axis when $ x = 5 $ as the $ x $ -intercept of the given line is $ 5 $ and we also know that on the $ x $ -axis, $ y = 0 $ . So, we have a point $ \left( {5,0} \right) $ which lies on the line $ 3x - 5y = 15 $ .
Similarly, the line crosses the $ y $ -axis when $ y = - 3 $ as the $ y $ -intercept of the given line is $ - 3 $ and we also know that on the $ y $ -axis, $ x = 0 $ . So, we have another point which lies on the line $ 3x - 5y = 15 $ as $ \left( {0, - 3} \right) $ .
Marking these two points on a graph and then joining the points through a line will give us the graphical representation of the line $ 3x - 5y = 15 $ .

Hence, this is the line $ 3x - 5y = 15 $ drawn on a graph.

Note: In an equation of the form $ y = mx + c $ , $ m $ represents the slope of the line and $ c $ represents the vertical intercept or $ y $ -intercept of the line as it is the value of $ y $ when $ x = 0 $ . Also, there is an alternative method to find the intercepts of a line equation. Convert the given line equation into intercept form of a line i.e., $ \dfrac{x}{a} + \dfrac{y}{b} = 1 $ , where $ a $ is the $ x $ -intercept and $ b $ is the $ y $ -intercept.