Question

The equation of the line with slope $7$ and $y-$ intercept $-5$

Answer 1

We have to find the equation of line, which is of the form y = mx + b. Here, b denotes the value of y-intercept and m denotes the slope. So, we can substitute the given data in the equation and then we will get the answer.

Complete step by step solution:
The “slope intercept form” of a line is represented by the below equation $y=mx+b$ , where $m$ represents slope and $b$ represents the $y-$ intercept. The $y-$ intercept is the value of $y$ when we have $x=0$. Slope is defined as the value that tells the steepness of line and is calculated using the formula $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. Thus, it can also be referred to as the ratio of the “vertical change” to the “horizontal change”.
We have been given that:
Slope of the line is $7$
The $y-$ intercept of the line is $-5$
$y-$ intercept is obtained at $x=0$.
Therefore, we can write the point that passes through the line as $\left( 0,-5 \right)$
The equation of the line is represented by:
$y=mx+b$ …(1)
Substituting the values of slope and $y-$ intercept in equation (1), we have:
$y=7\times x+\left( -5 \right)$ …(2)
We know that when there is a negative and positive sign and they are multiplied we get a negative number, we have $\left( + \right)\left( - \right)=-$
Therefore, equation (2) can be written as:
$y=7x-5$
Thus, this is the required equation of line when its slope is $7$ and $y-$ intercept is $-5$.

Note: We should keep in mind that $y-$ intercept is obtained when $x=0$, if by mistake we take $y=0$ the answer can come:
$y=mx+b$
$\begin{aligned} & y=m\times 0+b \\\ & y=b \\\ \end{aligned}$
Hence the answer obtained in this case is wrong so it should be kept in mind that we use the correct definition of $y-$ intercept.
Another method to solve this is that $y-$ intercept is obtained when $x=0$.
Point obtained is $\left( 0,-5 \right)$.
The equation of the line is obtained as:
$\begin{aligned} & y-{{y}_{1}}=m\left( x-{{x}_{1}} \right) \\\ & y-\left( -5 \right)=7\left( x-0 \right) \\\ & y+5=7x \\\ \end{aligned}$