Equation of the line with given x- and y-intercept

Question

What is the equation of the line with 5 and -3 as x- and y-intercept, respectively?

Is the answer $y = \frac{3}{5}x - 3$? I saw an example question online but I am not entirely sure with how you do this. Can anybody help?

Answer 1

Another way to look at this is by considering the equation of any straight line to be $$y=mx+c$$ where $m$ is the gradient of the line and $c$ is the y-intercept of the line.

We are given that the $y$-intercept is $-3$ so we now know that our line has the equation $$y=mx-3$$

There are then 2 ways of finding the gradient of our line. The first is to substitute in known values of $x$ and $y$ and then solve the resulting equation. We could pick the point $(5,0)$, which is the $x$-intercept. This would give us \begin{align}y&=mx-3\\ 0&=m\times 5-3\\ 3&=5m\\ m&=\frac 35\end{align}

meaning that our final equation is, as you have found, $$y=\frac35x-3$$

We can also calculate the gradient from two points, using the equation $$m=\frac{y_2-y_1}{x_2-x_1}$$

Our two points are $(x_1,y_1)=(5,0)$ and $(x_2,y_2)=(0,-3)$

Putting this into the equation gives us \begin{align}m&=\frac{-3-0}{0-5}\\ &=\frac{-3}{-5}\\ &=\frac35\end{align}

which also results in the equation $$y=\frac35x-3$$

Answer 2

There exists a general formula: if the $x$ and $y$-intercepts are $a\ne 0$ and $b\ne 0$ respectively, the line has equation $$\frac xa+\frac yb =1.$$ Apply it and rewrite it in the form $\; y=\dots$.

Answer 3

The equation of a line is $$y=ax+b$$

then you can find y-intercept if you let $x=0$ so here we have $b$ as y- intercept, also x-intercept which is indeed the only root of the equation can be derived by setting $y=0$, so for example here we have $$\frac{y-b}{a}=x$$

also your answer is true.

Answer 4

Derivation of Bernard's formula:

Start: $y=mx+b$, assume $a,b,m \not=0$.

1) $x=0$ gives the $y$-intercept:

$y=b$;

2) $y=0$ gives the $x$-intercept, say $a$.

$0=mx+b$; $a=-b/m$;

3) $y=mx+b=-(b/a)x+b$;

4) $y/b+x/a=1$