5-1/2r=3/4r-10

Solution for 5-1/2r=3/4r-10 equation:

5-1/2r=3/4r-10

We move all terms to the left:

5-1/2r-(3/4r-10)=0


Domain of the equation: 2r!=0

r!=0/2

r!=0

r∈R



Domain of the equation: 4r-10)!=0

r∈R


We get rid of parentheses

-1/2r-3/4r+10+5=0

We calculate fractions


(-4r)/8r^2+(-6r)/8r^2+10+5=0

We add all the numbers together, and all the variables

(-4r)/8r^2+(-6r)/8r^2+15=0

We multiply all the terms by the denominator


(-4r)+(-6r)+15*8r^2=0

Wy multiply elements

120r^2+(-4r)+(-6r)=0

We get rid of parentheses

120r^2-4r-6r=0

We add all the numbers together, and all the variables

120r^2-10r=0

a = 120; b = -10; c = 0;
Δ = b2-4ac
Δ = -102-4·120·0
Δ = 100
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{100}=10$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-10}{2*120}=\frac{0}{240}
=0
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+10}{2*120}=\frac{20}{240}
=1/12
$