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Write the equation of the line perpendicular to 5x-4y=1 that passes through the point (1,-6) in slope form AND in standard form.

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Answer:


\textsf{Slope form}: \quad y=-(4)/(5)x-(26)/(5)


\textsf{Standard form}: \quad 4x+5y=-26

Explanation:


\boxed{\begin{minipage}{6.3 cm}\underline{Slope-intercept form of a linear equation}\\\\$y=mx+b$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $b$ is the $y$-intercept.\\\end{minipage}}

Given equation:


5x-4y=1

Rewrite in slope-intercept form:


\implies 5x-4y+4y=1+4y


\implies 5x=1+4y


\implies 5x-1=1+4y-1


\implies 5x-1=4y


\implies (5)/(4)x-(1)/(4)=(4y)/(4)


\implies y=(5)/(4)x-(1)/(4)

Therefore, the slope of the line is ⁵/₄.

If two lines are perpendicular to each other, their slopes are negative reciprocals.

Therefore, the slope of the perpendicular line is -⁴/₅.

Substitute the found slope -⁴/₅ and given point (1, -6) into the slope-intercept formula and solve for b:


\implies -6=-(4)/(5)(1)+b


\implies -6=-(4)/(5)+b


\implies b=-(26)/(5)

Therefore, the equation of the perpendicular line in slope form is:


y=-(4)/(5)x-(26)/(5)


\boxed{\begin{minipage}{5.5 cm}\underline{Standard form of a linear equation}\\\\$Ax+By=C$\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are constants. \\ \phantom{ww}$\bullet$ $A$ must be positive.\\\end{minipage}}

Multiply both sides of the equation in slope form by 5:


\implies y \cdot 5=-(4)/(5)x\cdot 5-(26)/(5)\cdot 5


\implies 5y=-4x-26

Add 4x to both sides:


\implies 5y+4x=-4x-26+4x


\implies 4x+5y=-26

Therefore, the equation of the perpendicular line in standard form is:


4x+5y=-26

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