Question

How do you write an equation of the line with $x$-intercept 3 and $y$-intercept $- 2$?

Answer 1

Here, we need to find the equation of the given line. We will use the intercepts to find two points that lie on the line. Then, we will use the two points to find the change in $y$ and change in $x$, and hence, find the slope of the line. Finally, we will use the point-slope form of a line to find the required equation.

Formula Used:
The point slope form of a line is given by $y - {y_1} = m\left( {x - {x_1}} \right)$, where $\left( {{x_1},{y_1}} \right)$ is a point lying on the line, and $m$ is the slope of the line.

Complete step-by-step solution:
The $x$-intercept of the given line is 3.
This means that the line touches the $x$-axis at the point $\left( {3,0} \right)$.
The $y$-intercept of the given line is $- 2$.
This means that the line touches the $y$-axis at the point $\left( {0, - 2} \right)$.
Thus, we get the points $\left( {3,0} \right)$ and $\left( {0, - 2} \right)$ that lie on the given line.
Now, we will find the change in $y$ and change in $x$.
The change in $y$ from $\left( {3,0} \right)$ to $\left( {0, - 2} \right)$ $= - 2 - 0 = - 2$
The change in $x$ from $\left( {3,0} \right)$ to $\left( {0, - 2} \right)$ $= 0 - 3 = - 3$
The slope $m$ of a line is equal to the change in $y$, divided by the change in $x$.
Therefore, we get
$\begin{array}{l}m = \dfrac{{ - 2}}{{ - 3}}\\\ \Rightarrow m = \dfrac{2}{3}\end{array}$
Now, we will use the point slope form of a line.
Substituting 3 for ${x_1}$, 0 for ${y_1}$, and $\dfrac{2}{3}$ for $m$ in the point slope form of a line $y - {y_1} = m\left( {x - {x_1}} \right)$, we get
$\Rightarrow y - 0 = \dfrac{2}{3}\left( {x - 3} \right)$
Multiplying the terms using the distributive law of multiplication, we get
$\Rightarrow y = \dfrac{2}{3}x - 2$
Rewriting the equation, we get
$\Rightarrow \dfrac{{2x}}{3} - y = 2$
Dividing both side of the equation by 2, we get
$\Rightarrow \dfrac{x}{3} - \dfrac{y}{2} = 1$
The required equation of the given line is $\dfrac{x}{3} - \dfrac{y}{2} = 1$.

Note:
We have used the distributive law of multiplication in the solution to multiply $\dfrac{2}{3}$ by $\left( {x - 3} \right)$. The distributive law of multiplication states that $a\left( {b + c} \right) = a \cdot b + a \cdot c$.
We can also find the equation using the intercept form of a line.
The intercept form of a line is given by $\dfrac{x}{a} + \dfrac{y}{b} = 1$, where $a$ and $b$ are the $x$-intercept and $y$-intercept cut by the straight line respectively.
Substituting 3 for $a$ and $- 2$ for $b$ in the intercept form of a line, we get
$\begin{array}{l} \Rightarrow \dfrac{x}{3} + \dfrac{y}{{ - 2}} = 1\\\ \Rightarrow \dfrac{x}{3} - \dfrac{y}{2} = 1\end{array}$.