How do you find a standard form equation for the line with #(7,-4)# and perpendicular to the line whose equation is #x-7y-4=0#?

Question

Isaiah Anthony · Accepted Answer

#7x+y=45# #"the equation of a line in "color(blue)"standard form"# is. #color(red)(bar(ul(|color(white)(2/2)color(black)(Ax+By=C)color(white)(2/2)|)))# where A is a positive integer and B, C are integers. #• " given a line with slope m then the slope of a line"# #"perpendicular to it is"# #•color(white)(x)m_(color(red)"perpendicular")=-1/m# #"rearrange "x-7y-4=0" into "color(blue)"slope-intercept form"# #•color(white)(x)y=mx+b# #"where m is the slope and b the y-intercept"# #rArry=1/7x-4/7rArrm=1/7# #rArrm_(color(red)"perpendicular")=-1/(1/7)=-7# #rArry=-7x+blarr" partial equation"# #"to find b substitute "(7,-4)" into the partial equation"# #-4=-49+brArrb=45# #rArry=-7x+45larrcolor(red)" in slope-intercept form"# #rArr7x+y=45larrcolor(red)" in standard form"#

Eva Abramson · Answer

#7x+y=45#

The given equation is in standard form which is #ax +by=c#

but we need to know its slope, so change it into the form #y=mx+c#

#x-7y-4 =0" "rarr x-4 =7y" "rarr7y = x-4 #

#y=color(blue)(1/7)x -4/7#

#rarrm=1/7#

If lines are perpendicular then #m_1 xx m_2 = -1#

(One slope is the negative reciprocal of the other - flip and change the sign.)

The slope perpendicular to #1/7# is #-7#

Now we have a point #(7,-4)# and #m =-7# so substitute into the point-slope formula for a straight line.

#y-y_1 =m(x-x_1)#

#y-(-4) = -7(x-7)#

#y+4 = -7x+49" "larr# change to standard form:

#7x+y = 49-4#

#7x+y=45#

Eva Adamson · Answer

undefined To find the standard form equation for the line perpendicular to $x - 7y - 4 = 0$ and passing through the point (7, -4):

1. First, find the slope of the given line. Rewrite the equation in slope-intercept form, $y = mx + b$, where $m$ is the slope.
2. The slope of the perpendicular line is the negative reciprocal of the slope of the given line.
3. Use the point-slope form $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is the given point, to find the equation of the perpendicular line.
4. Convert the equation into standard form $Ax + By = C$.

Let's go through the steps:

1. $x - 7y - 4 = 0$ ⇒ $x = 7y + 4$
   Comparing with $y = mx + b$, the slope of the given line is $m = \frac{1}{7}$.

2. The slope of the perpendicular line is $m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{\frac{1}{7}} = -7$.

3. Using the point-slope form with the point (7, -4):
   $y - (-4) = -7(x - 7)$
   $y + 4 = -7x + 49$
   $y = -7x + 45$

4. Convert to standard form:
   $7x + y = 45$