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.