Question

Write an equation (a) in slope-intercept form and (b) in standard form for the line passing through (1,7) and perpendicular to 3 x + 7y = 1.a) The equation of the line in slope-intercept form is   enter your response here.write the equation in standard form also.

Answer 1

a.)

`y=-(3)/(7)x+(1)/(7)`

b.)

`7x-3y=-14`

Given:

`3x+7y=1`

To write the given equation to its slope-intercept form, we'll just have to write it in terms of y.

`\begin{gathered} 3x+7y=1 \\ 7y=-3x+1 \\ y=-(3)/(7)x+(1)/(7) \end{gathered}`

Now, perpendicular lines have the negative reciprocal of the other line.

`m_(\perp)=-(1)/(m)`

Since the slope of the given equation is -3/7,

`\begin{gathered} m_(\perp)=-(1)/(m) \\ =-(1)/(-(3)/(7)) \\ m=(7)/(3) \end{gathered}`

Now that we got the slope, we will substitute this to the following equation with the point (1,7)

`\begin{gathered} y=mx+b \\ 7=(7)/(3)(1)+b \\ 7=(7)/(3)+b \\ b=7-(7)/(3) \\ b=(14)/(3) \end{gathered}`

With this, we now know that the y-intercept of the equation that we are looking for is 14/3. Substituting the slope and y-intercept to the equation and we will get:

`\begin{gathered} y=mx+b \\ y=(7)/(3)x+(14)/(3) \end{gathered}`

Then, to write this in its standard form,

`\begin{gathered} y=(7)/(3)x+(14)/(3) \\ 3y=7x+14 \\ -7x+3y=14 \end{gathered}`

Since the first term cannot be negative, we will multiply the entire equation by -1. Therefore the answer for b is 7x - 3y = -14