Question

How do you write the Equation of a line that goes through \[(8,5)\] and is perpendicular to \[2x - y = 7\] in slope intercept form and standard form?

Answer 1

Hint: In this question, the slope can be found out by first finding the slope of the given line, then using the value of the product of slopes of two perpendicular lines. Then, the values of the given point can be inserted to find out the value of the constant, and thus, an equation can be formed.

Complete step by step solution:
First, we will obtain the slope of the required line.
Converting \[2x - y = 7\] to slope-intercept form,
 \[y = 2x + 7\]
On comparing it with the standard slope-intercept form ( \[y = mx + c\] ), we get the value of \[m\] as:
 \[m = 2\]
Thus, we obtain the slope of this line as 2.
Now, let the slope of this line be \[{m_1}\] , and the slope of the required line be \[{m_2}\] .
We have,
 \[{m_1} = 2\] and \[{m_2} \times {m_1} = - 1\] (Since the product of the slopes of two perpendicular lines Is -1)
From this, we get the slope of the required line to be \[\dfrac{{ - 1}}{2}\] .
Let the slope intercept form of our line be \[y = mx + c\] .
Now, as slope = \[\dfrac{{ - 1}}{2}\] , we replace the slope, and get the equation as
 \[y = \;\;\dfrac{{ - 1}}{2}x + c\]
We also know that this line passes through \[(8,5)\] , therefore the equation will be satisfied for \[x = 8\] and \[y = 5\] .
 \[
  5 = \;\;\dfrac{{ - 1}}{2} \times 8 + c \\
  c = 9 \\
 \]
Thus, the final equation generated in the slope-intercept form will be \[y = \;\;\dfrac{{ - 1}}{2} \times x + 9\] .
Converting this to the standard form, we get
 \[x + 2y = 18\]
Thus, the equation of a line that goes through \[(8,5)\] and is perpendicular to \[2x - y = 7\] can be written as follows:
Slope-Intercept form: \[y = \;\;\dfrac{{ - 1}}{2} \times x + 9\]
Standard form: \[x + 2y = 18\]

Note: The slope-intercept is the most “popular” form of writing a straight line. Many students find it useful, because of its simplicity. A student can easily describe the characteristics of the straight line without even looking at its graph, because the slope and y-intercept can easily be identified or read off from this form.
The standard form is simply another generalized way of writing an equation.