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Math  /  Algebra

Questionyc=−53y+23\frac{y}{c}=\frac{-5}{3 y+23}

Studdy Solution

STEP 1

1. We are given a rational equation involving the variable y y .
2. We need to solve for y y in terms of c c .

STEP 2

1. Cross-multiply to eliminate the fractions.
2. Simplify and solve the resulting equation for y y .

STEP 3

To eliminate the fractions, cross-multiply. This means multiplying the numerator of one fraction by the denominator of the other and setting the products equal:
y⋅(3y+23)=−5⋅c y \cdot (3y + 23) = -5 \cdot c

STEP 4

Expand the left side of the equation:
y⋅3y+y⋅23=−5c y \cdot 3y + y \cdot 23 = -5c 3y2+23y=−5c 3y^2 + 23y = -5c

STEP 5

Rearrange the equation to set it to a standard quadratic form:
3y2+23y+5c=0 3y^2 + 23y + 5c = 0

STEP 6

Solve the quadratic equation for y y using the quadratic formula, which is given by:
y=−b±b2−4ac2a y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
In this equation, a=3 a = 3 , b=23 b = 23 , and c=5c c = 5c :
y=−23±232−4×3×5c2×3 y = \frac{-23 \pm \sqrt{23^2 - 4 \times 3 \times 5c}}{2 \times 3}

STEP 7

Calculate the discriminant and simplify:
y=−23±529−60c6 y = \frac{-23 \pm \sqrt{529 - 60c}}{6}
The solution for y y in terms of c c is:
y=−23±529−60c6 y = \frac{-23 \pm \sqrt{529 - 60c}}{6}

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