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To write a equation of a line perpendicular to the given equation, we first need to determine its slope. Two lines are perpendicular when their slopes are negative reciprocals. This means that the product of the slopes of perpendicular lines is $-1.$
$m_{1}⋅m_{2}=-1 $
Note that the given equation is written in slope-intercept form.
$y=2x+9 $
In the above formula we can see that the slope is $2.$ By substituting this value for $m_{1}$ into our equation, we can solve for the slope of the perpendicular line, $m_{2}.$
Any perpendicular line to the given equation will have a slope of $-21 .$ Let us now write a partial equation in slope-intercept form for a perpendicular line to the given equation.
$y=-21 x+b $
By substituting the given point $(5,4)$ into this equation for $x$ and $y,$ respectively, we can solve for the $y-$intercept $b$ of the perpendicular line.
Now that we have the $y-$intercept, we can write the equation of the perpendicular line to $y=2x+9$ through the point $(5,4).$
$y=-21 x+213 $

$y=-21 x+b$

$4=-21 ⋅5+b$

Solve for $b$

MoveRightFacToNumOne$b1 ⋅a=ba $

$4=-25 +b$

AddEqn$LHS+25 =RHS+25 $

$4+25 =b$

RearrangeEqnRearrange equation

$b=4+25 $

NumberToFrac$a=22⋅a $

$b=22⋅4 +25 $

MultiplyMultiply

$b=28 +25 $

AddFracAdd fractions

$b=213 $