Question

What is the equation of line that passes through \[(4,7)\] and has a slope of $0.5$?

Answer 1

Hint: If ‘$m$ ’ is the slope of line and it passes through a point \[({x_1},{y_1})\], then its equation is written as: \[y - {y_1} = m(x - {x_1})\]
Given: A point \[(4,7)\] through which the line passes and its slope which is equal to $0.5$.
To find: Equation of line with slope .5 and passes through a point \[(4,7)\]

Complete step-by-step solution:
Step 1: From point slope form we know that the equation of line is given by the equation,
\[y - {y_1} = m(x - {x_1})\]
Now here it is given that,
\[{x_1} = 4,{y_1} = 7\& slope(m) = 0.5\]
Substituting the values in above equation of line, we get
 \[y - {y_1} = m(x - {x_1})\]
\[y - 7 = 0.5(x - 4)\]
This form is known as point slope form.
Step 2: rearranging the terms on both side of the above equation
We get,
\[ y - 7 = 0.5(x - 4) \]
\[\Rightarrow y = 7 + 0.5(x - 4) \]
Step 3: On further simplification, we get
$ y = 7 + 0.5(x - 4) $
$\Rightarrow y = 7 + 0.5x - 2 $
$\Rightarrow y = 0.5x + 5 $
The form of the above obtained equation i.e. \[y = 0.5x + 5\]is known as slope intercept form.
Hence, the equation of line that passes through a point\[(4,7)\]and has a slope equal to$0.5$ is,
\[y - 7 = 0.5(x - 4)\] Or \[y = 0.5x + 5\]
Point Slope Form $\;\;\;\;\;$ Slope Intercept Form
Additional information:
> If a and b are the intercepts made by a line on the axes of x and y, its equation is written as:
 \[\dfrac{x}{a} + \dfrac{y}{b} = 1\], this form is known as the Intercept form.
> Equation of line passing through two points \[({x_1},{y_1})\& ({x_2},{y_2})\] is written as:
\[y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_1})\], this form is known as two point form.

Note: These minor details should be noted
$>$ Equation of a line parallel to x-axis at a distance \[a\] is \[y = a\] or \[y = - a\]
$>$ Equation of x-axis is \[y = 0\]
$>$ Equation of line parallel to y-axis at a distance \[b\] is \[x = b\] or \[x = - b\]
$>$ Equation of y-axis is \[x = 0\]